Yes I have and do all the time.
That being said I have a JD 2520 compact tractor. It is 26 horse power. Approximately the same footprint as a side by side or a golf cart. . I use a 3pt hitch trailer mover that I built, but they are readily available to buy on eBay.
I only know JD model numbers, but there are plenty of other brands that are great also, and some are a bit cheaper, but surprisingly the green paint is usually not the most expensive. For a JD model the smallest that would be really good for moving a camper sized trailer is probably the 2025R and it will have around 1100 pounds of 3 point hitch lift capacity.
That being said, I love moving my trailers around with my little tractor. Since it is on the 3pt hitch I am able to lift it, so I just back up to the trailer, lift the ball up using the hitch height control, and then drive away. So getting off to crank the jack. It is also very maneuverable to get into tight areas. The only downfall is it is very hard to see behind to back up, as you are sitting centered with the trailer and don't have mirrors. To back up into anything but an open area you would need either a spotter or mount a back up camera on the back of the traile and a monitor on the tractor hood.
I also love love love mowing with my tractor. Being a diesel it cuts so fast and has the torque to not bog down I. Heavy or wet grass. The only downfall with mowing is it has a roll bar, so you can't get far under overhanging tree branches.
Brakes aren't a concern moving trailers around at yard speeds on relatively flat ground. The tractor has decent brakes and if you were in a panic you could drop the hitch down onto the jack. If you did have hills and wanted brakes it wouldn't be hard to mount a brake controller on the tractor, and apply the brakes through the slide switch on the controller.
3pt trailer movers and moving trailers with compact tractors is very very common on the compact tractor forum I am a part of.