So the only way I can think of is to add a 1Ohm resistor between the LED and R1, and then use two probes on that added resistor and the MATH function of the oscilloscope to subtract the values.
You already have a 180 Ohm current sense resistor, giving you 180x larger voltage (and thus better sensitivity!) than a 1 Ohm current sense resistor would. Zero changes to the circuit needed.
You can (and should!) use the preset store/recall of the scope to make setting up such a measurement take a fraction of a minute, in case you worried that bringing up the Math menu and setting it up every time is too much hassle (it's not once you've used the scope enough - but do use the presets, they are handy!).
This seems pretty complicated just to measure a single current.
For reference: in a professional setting, I just grab a DC hall current probe, clip it on a wire in the circuit, hook output to the oscilloscope, and call it a day.
Such probes, with good specs, go for $1k+. So you're essentially claiming that the value you get out of connecting just two cheap voltage probes and using a MATH function on the scope - that you paid for! - is not a good deal or is "too cumbersome"? It's an excellent deal! Be happy you can do this measurement so easily!
But, if you think that just one voltage probe will really save your day, and assuming that your circuit is not isolated from the mains ground, then adding a diagnostic current mirror is the next best thing.
A current mirror can make a copy of the current that is ground-referenced:
simulate this circuit – Schematic created using CircuitLab
Q2,Q3,Q4 form a current mirror that copies the current from its left branch to the right branch. R3 is a load that then converts that current into a voltage: 1V per each 10mA of load. The measurement error is about 3%. D2,R3,R5 replicate the load on the left branch of the current mirror, and keep the collector potentials of Q2 and Q4 similar. This keeps the accuracy of this approach reasonable in spite of its simplicity. D1 and D2 should be the same type.
R3+R4+C1 form a low-pass filter with time constant of 11ms. The transient response plot below shows the output setting; the average current is about 4.7mA, the PWM is 50% @ 10kHz.
If you want a better current mirror, you'll need a couple more transistors. The measurement error is well below 1%, and is mostly determined by the accuracy of R3, as long as the transistors are not terribly mismatched.
simulate this circuit
This latter current mirror design comes from “Current mirror circuit with accurate mirror gain for low β transistors” by Chen, Whiteside and Geiger, and performs admirably with discrete transistors.
Q8's collector can be alternatively connected to ground, instead of the base. This alternative configuration is best evaluated experimentally, and chosen if it's more accurate.
I often leave such diagnostic capabilities in-circuit for small production runs, since when there's a problem, they pay for themselves handsomely. In non-trivial circuits, the cost of a couple dozen extra surface mount transistors is essentially nil, and there never are shortages of those parts, vs. let's say a current sense amplifier chip (never mind its cost!).