You might find gear + 2 overwhelms most boats, but self-support IKing is a great way to see rivers. You might want 2 boats though.
I've been pretty impressed with the tandem outfitter as a gear hauler that still paddles better than the narrower tandem lynxes or strikes. But I've never tried to put 2 + gear in one.
a single IK, with 1 person and gear, is going to boat a lot more responsibe though. I prefer burly drybags, watershed or drysuit zippered, rather than roll top. Keep your load tight and low.
The IK is a couch, cot, table, hammock, etc ....
I'd consider the upper san juan to mexican hat a good first overnighter, it's short enough you could even day trip it in light boats, then come back with gear.
SW has some real gems. The Gila through the Mazatzal wilderness is quite lovely, and the Verde is another commonly run small river. You can IK the Salt at silly low flows and have it to yourself. (Watch out for Black Rock at ELF!) Then there's all the short-season oddball creek stuff, not all of which is hard whitewater, like the Muddy Chute.
Despite a lot of water rated no more than III, though, I'll comment that I've found wilderness rivers in the SW to be a little more demanding than the rating might suggest. Not so much in technical difficulty, but more in terms of river visibility. There is some kind of rush or grass along the Gila, for example, that creates blind, deep, fast channels. Wood is less rare than you'd think on desert rivers and would be quite dangerous in some of those spots.
Just be aware when on things that are not often rafted that you might face creeking conditions of blind wood, portages, sieved-out drops, etc.
I've got some boating family in the area, drop me a PM if you'd like contact info.