Don't think a jack is needed
Assuming a double axle trailer, roll the trailer with the still good tire up on a 4x4 block you probably have (at least I do) to hold the car when loading the racecar. That should give enough clearance on the other axle to change the bum tire.
Tim
Trailer Tire Pressure Monitoring
I have heard of some people using tire pressure monitor systems designed for trailers that can give in cab readings. I think this would at least help with the slow leak, heat building scenario.
Thanks.
Ken