You won't need to raise both shells all the way up, but you will need about 6" of rise. You must absolutely NOT use the corner levelers to raise one side (the side with the bad tire) higher than the other -- the TM will slip downhill on those levelers, sideways, if you don't keep it perfectly level. You will need to disconnect the hitch. You should leave the TM connected at the hitch ball and set the parking brakes on the tow vehicle. First, if you use a WDH, you will need to loosen or disconnect your weight distribution tension bars. In order to remove the wheel well cover, you will need a #2 square drive screwdriver in the tow vehicle. You should already have a torque wrench, with sockets for both the TM and the tow vehicle, and you should also have a 6" long extension for that wrench and socket set - you need the extension to to reach into the TM wheel hub from outside of the TM lower body panels. You should also have a decent 12V air pump in the tow vehicle - and that pump must be able to able push tires up to the pressure values which trailer tires require.
If you use jack stands and a different jack, you can raise one side higher than the other. But you need to be sure to put down blocks, in order to prevent sliding, and you need to have good emergency lighting if you are doing the change on a roadway shoulder.
First get the spare on the ground, and re-inflate using the 12V air pump which you keep in the tow vehicle. Then loosen the lug nuts on the mounted wheel partway (while the wheel and damaged tire are still on the ground, because it is much harder to loosen fully tightened lug nuts after the wheel has been raised to spin in free air.) Set at least one of your jack stands and begin raising the trailer. NEVER GO UNDERNEATH THE TRAILER OR FRAME EDGE WHILE IT HAS BEEN RAISED. While raising the TM, raise your jack stands an additional notch as soon you have room to move their locking pins for more height, or add blocks underneath them.
After raising enough, check that the wheel with the bad tire is capable of rotating on the hub - because a a "burned bearing" type of problem will quickly destroy your spare, just as quickly as it destroyed your original tire. The remove the nuts and wheel with the bad tire.
Next check for damage to the hub, brakes, and brake wires, then use a dry cloth to clean the "bolts" of the wheel hub to be both dry and fairly clean without dirt, and without any grease or oil residue). Then put the spare in place, hand tighten the lug nuts and make sure that the wheel is vertical on the wheel hub. Then tighten the lug nuts to about 40-50 foot-pounds in an alternating pattern.
Lower the tire to the ground - not all the way, but enough the re-torque the lug nuts without causing the wheel to spin. First to 80 lbs (in an alternating pattern) then up to specification - about 100 lbs IIRC. 110 is probably OK and unlikely to damage the hub threads, but do not go higher.
Put the bad wheel+ tire up into the carrier. Re-hitch the trailer, then check that all of your blocks, jack stands, jacks, and leveler jacks are raised. The start the truck and test the brakes before driving.