The TPS63020 has a maximum internal switch current limit of 4A (max 4.5A) so this will correspond to the maximum peak current of the inductor, and the minimum saturation current of the inductor (anything >5A will work nicely). It appears that the cavity is quite large, so you should be able to substitute the inductor with anything that will fit. Because the total power is not very high, I would not be too concerned about the need to re-pot the inductor - it floating in the cavity will be fine.
The exact choice of the inductor is a little too complex to discuss over here, but in general, it is a trade-off between higher inductance (and lower ripple current) and cost / size and DCR (where higher inductance values typically have more DCR for a given size). In this case, my guess is that it is unlikely that the feedback loop is too sensitive to the inductance value, and something like 1.0, 1.2 or 1.5uH will work. I would pick something that would have the lowest DCR for the size, to improve overall efficiency (not a big difference but why not).
It looks like the Coilcraft XEL4030-152ME_ or XEL4030-122ME_ should fit, and is likely to be a small upgrade over the stock inductor (likely lower DCR). But you have to purchase this from www.coilcraft.com directly. You may be able to 'request free samples' and just pay for shipping (or maybe it's free).
Otherwise, the previous Bourns inductor I linked appears to be a decent replacement as well (you just need to measure the cavity to make sure it fits in both depth, height, and width).
To replace the inductor, I would use needle tweezers and a regular soldering iron to remove any broken fragments of the old, broken inductor.
Then I'd de-tin the pads using solderwick. Most likely, ZL uses a lead-free process - lead-free solder doesn't flow as easily and has a higher melting temperature, so it's more difficult to rework. Personally I like Solderwick products for this. Once free of solder, I'd reapply a small amount of leaded solder using a soldering iron so it just covers the pads. Use less solder than you think you need!
At this point, the best thing to do is to clean the pads with isopropyl alcohol and a small brush to get rid of old flux and residue. Then I typically reapply some flux (I use this, any of them will work well: https://www.chipquik.com/store/index.php?cPath=300), and you can use hot air to reflow the new inductor on.
Note: You may be able to find another inductor with exposed pads, which you can solder directly to the pad with a soldering iron instead using exposed flanges. One example (if it fits) is this: https://www.digikey.com/en/products/detail/vishay-dale/IHLP2020CZER1R5M11/2025075
I wouldn't worry about the other components on board. Just make sure the airflow is not too strong to blow the caps off. The LED and MCU are on the other side so it shouldn't be too much of a problem, but I typically try not to get the hot air too near the MCU (may mess up the flash memory) or the LED (may weaken the silicone dome joint to the substrate).
The process is the same for your LED swap. It would be nice to see a 4500K 9080CRI Nichia inside, but I'm sure you already have something nice planned.
Good luck!