If you put a short length socket on the lugnut,can you turn the lug nut by hand or do you have to use a ratchet to turn it?? The purpose of using a short length socket is so you can look through the square hole (that the ratchet fits info) to see if the stud is turning or not.If you can turn the lugnut by hand,turn it and see if its just the lug nut spinning or the wheel stud spinning too?? If the wheel stud is spinning too,the splined portion under the head of the wheel stud could have stripped or the wheel stud could have backed out of the axle just enough to where the splines are no longer digging into the axle (which is what keeps the studs from turning) If you look closely at the remaining 3 studs,does the one youre working on look like its coming through the rim the same distance as the other 3?? If not,that verifies the stud has backed out.
If youre 100% positive the stud is locked into the axle tightly and the lug nut itself is the only thing turning,you could try this::
Get a buddy or your wife to help if needed.Position the rim to where you can get the most room for leverage to pry against the backside of the rim/tire,in an area closest to the stripped lugnut.In otherwords,youre gonna have somebody slide a long pipe between the backside of the rim/tire & the metal wheel well.They can start prying outwards on the rim/tire now (directly in line with the stripped lugnut) while you simultaneously try removing the lugnut.Youll have a much better chance at removing the lugnut,if you use an air compressor powered impact gun,because of the torque and speed an impact gun applies to the nut.If you can jar the nut loose (with the torque and speed of an impact gun) while applying pressure to the backside of the lugnut,it might be enough to get the lugnut off.
Theres actually a tool made called a "Nut Splitter" thats sold at your local part store.It works by splitting the nut from its side,but I dont think it would work on your nut because of how much its recessed into the rim.If you could get to at least a 1/2 of the nuts side,you could split it using the tool then split the remaining part of the nut by using a hammer & chisel.Unfortunately, it looks like 85% of your lugnut is surrounded by the rim,which would make it impossible.
One other method would give you alot more room to work in,but I dont know if it would help removal or not plus it would be added work too.If the brake shoes are loose enough from the drums,you could drain the rearend housing,remove its cover,remove the axle C-clip and extract the axle,rim,tire & brake drum in one assembly.That would allow you to set it on a work bench for more removal options or you could take it to a machine/brake shop to see if they can remove it.