From years of pulling trailers of all types and sizes, I've never rotated tires. The point of rotating tires is to adjust to uneven wear due to steering, camber, toe, etc. If your trailer has a camber issue, its related to the axle or the hub. If its the axle, then it technically should be the same on other side, so rotating makes little difference. If its the hub, stop what you're doing and fix the hub. If it has a toe issue, that definitely should be fixed. Understood, though, that the OP is simply asking out of curiosity.
Regarding the spare, I use the same spare for multiple trailers, so rotating it into the mix isn't really a good option for me.