ACK your dog is adoreable. I don't know why, but recently I've been noticing those types of puppies and their little pushed in schnozes and I just want to hug them and feed them all bacon.
I basically lived in something like the Asolo tent (but 1 person and lighter weight - strictly summer here) for almost two years. It was comfy in the heat, and that style was a cinch to put together and take down extremely quickly (midnight sandstorms, flash floods, oops we camped in the middle of a road, etc - not problems you're likely to encounter here with a little better planning that I managed)
Our dog runs marathons in her sleep, we couldn't swing anything less than a three person tent and even that's a stretch (but she's ~55-60#). 15# is large cat size, but I'd still make sure that whatever it is has lovely ventillation options. If rain happens, you'll be glad for a vestibule, pup aside. I second the footprint recommendation (but usually bring a tarp anyway, j.i.c.).
None of that is bike-specific, on that I've got nothing. (Please post what you end up with, and the results, if you could?)
Just had a thought - if you're biking with the pup, that means maybe she'll have a trailer? She could hang out in the trailer at night while the people get the tent? Car camping with dogs, the pups stayed inside the car (and stayed drier than all the people through storms), so maybe the bike equivalent is a nice waterproof trailer pulled alongside? I could see it being not an option, but I figured I'd mention it.