as others have said, you aren't going to be able to pull much with only a 2000lbs. towing capacity.
THAT'S your biggest hurdle, your tow vehicle, not whether you get a bathroom or not.
you really need to start looking at what trailers weigh.
by your statement of having a bathroom or not, shows you don't seem to have a grasp on what trailers weigh.
only the smallest of popups would be under 2000lbs. there might be some teardrop trailers under that.
you really don't have much to choose from if you stay with that Jeep.
and none of them are appropriate for full-timing, especially with a large dog.