The rule of thumb is: the absolute value of a single resistor, no matter how good its tempco, can be handily beaten by the ratio of resistances of two or more much cheaper resistors.
Precision applications where an absolute value of a resistor is of essence indeed require expensive parts and ovenizing.
But nobody forces you to do it this way. The current source in LT3045 is a convenience for less critical applications. It is designed to be better than reasonably priced resistors, and nothing more than that. It basically lets you assume that if you select a typical metal film resistor, the I_SET tempco can be ignored. But this chip is simply not designed to use ultra-precise external reference resistors: this architecture simply doesn't make much sense then.
You shouldn't be using it when temperature stability you require is of essence. The SET terminal can be driven from a voltage source. There are no guarantees that the LT3045's current reference is any more stable than the rather expensive resistor you chose.
The tempco of the current source isn't even specified (sic!), but it's somewhere in the 50ppm/K ballpark around 35C, looking at the typical I_SET vs. temperature graph.
Also remember that resistor tempco may be specified over a much narrower temperature range than you'll get by touching the resistor with a soldering iron tip. Never mind that such precision resistors are only meant to provide specified tempco when they themselves are isothermal with the substrate they are on. Touching them with a soldering iron after the board is assembled is a surefire way to shift their value due to thermomechanical hysteresis, and is a rather destructive thing to even contemplate. Do not test precision resistors' tempco that way! If you contact the resistor manufacturer's application engineering, they'll probably tell you as much. Getting the tempco they specify requires some care, and abrupt temperature changes is definitely not on the path to long-term stability.
It is way, way easier to make a voltage source to your specs using an external voltage reference and a resistor network, than using a single ultra-expensive resistor and an oven. Especially that the current source is not as good as many even semi-competent voltage references would be.
For the price of the resistor you've shown, you can get a voltage reference with a tempco an order of magnitude better than that resistor's in practice, and that's without trying super hard either.
You haven't mentioned what the output voltage you need is, so let's assume it's 3V. With a 10ppm/K resistor, the voltage will drift 30uV/K. That's not a particularly impressive performance from 30 quid worth of parts! And that's assuming no drift from LT3045. And that's a wrong assumption. LT3045 is not 10ppm/K good. Not unless you'd select them for that tempco within the operating temperature range in your application.
You could probably get similar ballpark performance if you selected from a small batch of LT6650 parts for low drift around 35C, and used that to generate the reference voltage. LT6650 can be configured as a shunt regulator working from the 100uA current source, so the whole thing could be connected directly between SET and GND. Similar performance for 1/10th the cost.
With some selection, perhaps TL4050C could offer similar temperature stability.
If the voltage you need is covered by standard voltage references, then for a couple quid you can get references that go down to a couple ppm/K guaranteed, and will handily outperform the thermal errors within LT3045. At that point, you'll need an external DC servo that generates a voltage setpoint for the SET terminal, since the thermal drift of the internal error amplifier within LT3045 won't be a match for voltage references in this class.
You'll also find out that very likely the voltage distribution on the PCB, and the assorted thermal voltages in other circuitry, will be large contributors to the effective voltage tempco, even though the regulator won't be of fault.
I wonder why you need such a precise absolute voltage anyway? Most measurements can be made ratiometric versus a reference voltage, and then the absolute value of the reference voltage doesn't matter. If your application uses a sensing bridge, like a strain gage bridge, thermistor bridge, etc. - trying to keep the voltage reference to a low tempco is entirely pointless, since ultimately the A/D converter will be just dividing this voltage reference down and using it for reference. The value anywhere within 1% of the desired one will be plenty good.
Focusing on just this single resistor is missing the forest for the trees. In order to make any guesses as to how salvageable the whole design is, it'd help to have an overview of the system design for the whole thing. It may even be that the voltage doesn't matter, e.g. if the customer doesn't understand the ratiometric nature of certain common measurements. I have no idea how sophisticated the customer is, but such snafus aren't unheard of. Sometimes the customer just cares that the output from the ADC is stable, and they don't understand that this may or may not require absolute-value stable system voltages... Never mind that around 10ppm/K things kinda get hard, and tempcos stack up. Everything in the system that deals with absolute voltages would have to be much better than 10ppm/K in order for the overall system performance to be 10ppm/K.