Probably no pop-ups, as that weight limit, and the trailer frontal area restrictions you've not yet learned about, puts your choices in the range of the smallest available folding tent trailers, e.g. those marketed for towing behind motorcycles and small cars.
Smallest pop-ups today are the cold-water models (like Jayco's Sport and its Starcraft twin). In the eight-foot box, these are usually around 1300-1500 pounds lightly loaded (and now list around $8000 to $9000 new). Frontal area is about 35 sq ft, typically 7 feet wide by 5 feet high.
You can find these used, because these small simple models go back to the beginning of pop-ups, and all the pop-up manufacturers continued making at least one 8 foot and one 10 foot box in this lightweight category.
In the tent trailer category you'll find the
Sylvan Sport Go, and the
LivinLite Quicksilver to be the premium models, the Go approaching your weight limit with any sort of load, while the Quicksilver 6.0 is at about 600 pounds empty and would fit you and your dog.
The prices on these, however, are nowhere near your budget. Costco once sold a lower cost tent trailer kit based on a small utility trailer, under the
Lifetime brand. The kit price was $1999. The manufacturer has pulled this product from the market, but you might find it used.
Teardrops? The smallest, lightest models from T@B are around 1600-1800 pounds, the GVWR just under 3000. The market for these is small and mid-size SUVs which often have 3000-5000 pound tow capacity and can handle 50-60 sq ft trailer frontal area.
Besides the tent camping option, you could look for destinations and stops that have camping cabins. Except for high demand locations, these are usually about half the price of a nice motel room (but maybe not less than a not-so-nice room). You bring your own bedding and cooking gear, the cabins usually have power and running water, but not always toilet and shower facilities, those are shared on the campground.
Bringing a dog does work against the hotel/motel option, so I can understand the interest in bringing your own camping trailer.
