It occurred to me, does the air conditioner shroud account for 4" of height? Perhaps, removing the shroud may get you enough loss of height to get the trailer in the barn.
Any cheap jackstands X 4 will work for your purpose of getting the wheels off the ground (regarding weight). FWIW, the smallest stands I have are 10 3/4" from grade to inside of saddle. That will raise the overall height of the trailer if placed under the spring shackles. Don't know if that will work because ceiling truss height of your building is unknown.
An alternative would be to remove the 12" wheels and lower onto timbers for storage.