Just to amplify a bit on that, the spare tire well got deliberately larger at TS60001, when they went to 165 tires. Every 165 I've tried has been an extremely tight fit in the earlier well, if it fits at all. I used to keep a rope tied around the tire, and it partially deflated, just so I could get it back out.
Like Geo, I eventually went with a "compact spare", $15 at the local junkyard. Another advantage, the compact spares are a different construction; treadwear is very short but they don't "age out" like modern radials do. So they last essentially forever as a spare tire.
Just for clarity, what I was suggesting was to pull the hub and half shaft as an assembly; which is much easier. Turn the lock tabs, undo 6 bolts, and it slides right out.
Might be a good time to consider replacing the oil seal inside the housing. It may get damaged when you remove the axle and it's a PITA when they leak (gear oil ruins the brake linings).
Keep a suitable nut handy, so you can put one of the bolts back through the backing plate and shims. That way, you don't have to open the hydraulic system. I had the shoes off for this photo (mine had to be replaced anyway), but they can stay in place.
When you get it back together (but before turning the new lock tabs), it's best to recheck the end float. Should be OK, but sometimes it's not.
HF sells a clamping dial indicator that I found to be adequate for the task. I did have to add a flat washer to get the locking mechanism to work, though.
https://www.harborfreight.com/clamping-dial-indicator-63656.html