You could get all of these (plus the dogs' packs) into a 3-person Tarptent Rainshadow 2 (42 oz. if you use trekking poles; single wall; $265), or a 4-person Tarptent Hogback (65 oz.with poles included; double wall). The second is a little over your budget at $375. Unlike most American double-wall tents, you can pitch the fly first and the inner tent underneath it, thus keeping the inner tent dry in a downpour.

tarptent.com

I have the Rainshadow 2. (The "2" is the version number; it's actually a generously-sized 3-person tent.) It will hold three humans plus one dog (at the foot) with lots of room; two humans plus two dogs will have even more room (you could put one dog on each side of the tent, with the two humans in the middle)! I have no experience with the Hogback, which was new last spring.