I'm not aware of such a failure mechanism related to (coreless) Hall effect sensors, in and of themselves.
What comes to mind:
1. Fusing, as in, you've blown up the sensing element, or otherwise damaged the sensor from overheating.
This applies to pre-made sensors, with inductive loop included. The rating would be easy to exceed with this kind of ratio of ratings (i.e., in relation to the referenced thread, say 100A peak through a 15A-rated sensor). But there likely exists a pulse duration for which such currents can be reached, without damaging the device, so it's within the realm of possibility, and worth noting.
2. Excessive EMF, would be exceeding the CMRR of the sensor (due to voltage drop across the inductive link itself, or surrounding wiring), and either disturbing the reading (the sensor may return to normal after a settling time), or inducing too much voltage in the sensor itself, destroying microscopic wiring, or overheating the chip in an instant.
EMF is proportional to rate of change \$dI/dt\$, so this would be something of a limiting value in the other direction: too fast and you encounter this failure mode, too slow and you encounter the other. There exists a slope between these limits, where some maximum peak current can be withstood, reached by a pulse rising at the given \$dI/dt\$, and terminating before the element is damaged.
For cored types, there is no integral link, so limit #1 goes away (it would be using your own wiring instead, assumed of adequate size to handle whatever current you're putting through it), but limit #2 does remain (there's still some EMF transmitted through the core). We also have two additional considerations: absolute flux density, and hysteresis.
3. Electronic components, in general, malfunction in very intense fields. This isn't much of a problem for coreless sensors, because to get such fields (multiple tesla) around a wire link, would require massive surge currents to approach -- fractional lightning-strike currents, say (and then you probably fail from limit #1 or #2). But with a core, field from the sensed wire becomes concentrated, and this flux density needs to stay within limits.
What the limits are, isn't clear, or often documented. Most components are fine with fairly intense fields (100s mT, perhaps?). Some may even remain nominally operational up to multiple T. It's not hard, I think, to remain operational at fields of 5 or 10T, at least for some kinds of components, and this rules out a heck of a lot as you need either a very strong electromagnet indeed, or a superconducting one, to reach such field strengths. (Well, or a Halbach array, but that's also a fairly contrived and special thing.)
Magnetic cores exhibit saturation, where flux density can only rise so high for a given magnetization, and the rate \$B/H = \mu\$ decreases above some point; that is, \$B/H = \mu(H)\$ is a dependent and decreasing function. This is somewhat of an advantage here, as it limits this failure mode: while it doesn't affect rate (limit #2), it does limit the peak value.
Saturation flux density depends on magnetization, material and temperature. Ferrites are generally on the low side (0.3-0.45T), and iron alloys can go fairly high (1.5-1.8T, sometimes even 2T). Intermediate values can be achieved by varying the cross-section of the core: if say 80% of the core saturates, it's basically not there, and then some wider pole pieces/faces flanking the Hall effect sensor can cause the field to spread out, lowering it around the sensor. Conversely, sensitivity can be increased by pinching the core down around the sensor; this has the drawback that, the pinched section saturates progressively, lengthwise, so the response will be nonlinear (a softer limiting function) as it saturates.
4. Hysteresis, is a history-sensitive offset in the reading. Various conditions can affect the resting or residual level in a core, including magnetization, orientation in surrounding fields (including Earth's magnetic field), and even mechanical stress (simply being bumped around a bit). It's an intrinsic property of magnetic cores; "soft" or low-loss materials strive for low hysteresis, but some inevitably remains, and for sensitive sensors, this can be an annoyance.
Hysteresis is particularly important for DC readings, where the offset can't simply be ignored, but must be eliminated (degaussing?) or compensated (trimmed out, auto-zeroed, etc.). A typical circumstance is, using a DC-capable clip-on type current probe: when the core is opened, it's exposed to ambient fields, and then when slapped shut, it's stressed some. These tools have a prominent thumbwheel adjustment on them, so that the user can place the probe, turn off the circuit under test and zero the reading, and then take measurements.
How much any of these limits matter, depends on the materials, geometry, and sensor, in question. I would suggest working closely with a manufacturer to determine what their sensors are capable of, and whether any of these limits need to be designed around, or if any apply at all.
Offhand, I suspect your imagined application is fine:
A ferrite core will easily saturate at cranking currents (an ungapped ferrite of 2-3cm size typically saturates at 10At or so; with a chip-thickness airgap, perhaps 100At), which means plenty of signal strength for the low-level detection desired, and, ferrite is probably preferred over laminated iron to keep the flux density low. You don't need tremendous dynamic range, either.
dI/dt or EMF can be limited by placing a shield over the sensor; if solid foil is used, both common-mode voltage and rate-of-change can be limited. If rate-of-change should not be limited, then a slotted foil can be used for CM shielding (so that it doesn't act as a shorted turn, but still largely covers the sensor to absorb direct electric fields).
Hysteresis is also reduced by the airgap, roughly in proportion to its relative size (i.e., consider the (solid, whole) core as being an ideal magnetic path plus an intrinsic air gap of \$l_e / \mu_r\$; compare physical airgap to this figure). For typical dimensions, and common ferrite materials, this will probably be in the 10s of mA equivalent range. You can look up hysteresis for typical materials by locating the datasheet; the B-H curve is usually given. Special values have names: where the hysteresis loop crosses the vertical (B) axis, is remenance (remaining flux density at zero magnetization); horizontal (H) axis, coercion (how much magnetization is required to undo the remenance, or "coerce" it into magnetizing again). Again, smaller values of both are preferable.
Again, these are all very wishy-washy things, which will have to be concretized with a specific design, and then one can consider design variations motivated by the above reasoning.
 About dynamic range, by the way: the noise floor is most likely limited by the sensor itself. The Hall effect tends to be quite modest, requiring a lot of onboard gain to get a reasonable signal level, and this leaves a fairly generous noise floor. There is also Barkhausen noise, particularly due to the change of magnetic field -- usually explained as the stick-slip motion of magnetic domains across grain boundaries and pinning defects. Probably the noise limit will be some 10s of mA, adequate to positively detect fractional amperes. Filtering will suffice to reduce both noise sources, at least give or take whatever response time you require.
There are alternatives, as well:
Don't pooh-pooh the shunt resistor. With a very accurate current-sense amplifier (auto-zero or chopper types with ~µV input offset are readily available), both precision and accuracy are possible such that -- well, very accurate metering for battery charge can be done, for example. Even at low load currents, or large peak-average ratios, where tiny systematic errors are integrated over time into cumulative charge errors.
The compensated or nulling current transformer can have extremely low frequency response, by canceling out incident EMF through feedback action. They still do not go to DC however, which is probably not adequate for this case; it might be a consideration for pulsed operation, where a regular return to zero affords a calibration/reset opportunity.
Magnetic saturation can be used to advantage. By using a pair of cores, and driving them alternately into saturation, the difference (saturation flux) can be measured, and thus the load current; or instead of a direct proportional reading, it can be nulled via feedback (through a cancellation winding) giving a very linear result (i.e. not dependent on core characteristics as the cores operate over a consistent section of B-H curve). These are DC capable, and have been used for metering purposes (i.e. accuracy sufficient for electrical billing).
We might also imagine some circuits that could solve the problem, though I think it's stretching the imagination a bit for this particular case. Consider the humble series diode in parallel with a resistor: when load current is small, it flows through the resistor, and can be measured at reasonable accuracy with modest means; at high currents, the diode turns on, limiting voltage drop. Why it doesn't apply here: you can't really find diodes with selectable voltage drop; schottky have the least, and you aren't going to find less than 0.7V or so at this kind of peak current. Out of a 12V supply, that's a fairly noticeable efficiency hit. Or a synthetic diode could be made with MOSFETs and active circuitry, but idle current would likely suffer, and the circuit obviously becomes far more complex.