My first choice would be Dichondra (D. micrantha). In full sun it needs no mowing, though it will need twice-weekly watering and monthly feeding. If the pups use it as a bathroom, it will suffer more damage from urine than bermudagrass would.
If the area has some shade during the hottest part of the day--noon to 4:00 pm--you could use the groundcover thymes, though there will be more risk of bee stings when they bloom.
Myoporum (M. parvifolium or M. brevifolium) are a possibility that can grow in full sun, and is drought tolerant. They don't take traffic well, though, and they have the same bee problem when they bloom.
Hope these help! : )

Thanks so much for your suggestions. I really like the idea of thymes . . . I grow some in Oregon in sunny, sloping areas where I can keep it from drowning but I never would have thought about using it here! Thanks!

Just remember to supply some shade during the hottest part of the day for the traditional groundcover thymes (creeping-, mother-of-, woolly-, etc.) or they won't survive the summer. Caraway thyme and common thyme can take full sun, but they don't make as dense and traffic tolerant a cover.
I just remembered that Korean grass is a possibility, too, but the low humidity here makes it somewhat more brown tinged than you probably remember.
Happy hunting! : ])

Phyla nodiflora, Frog-Fruit, is a good one for using less water (once established) in full sun but it does also draw the bees. Is a wonderful spreader once it gets going and takes traffic very well. Will be hard to find though, mine came from Shady Way in east Mesa, I haven't seen it elsewhere and I haven't seen it there since but it would be worth a call to see.

Is there anything wrong with using the buffalo grass for dogs?

I've never heard of a problem, but it may be hard to find the right kinds of buffalo grass for the Valley. Queen Creek has a little more flex in that area than we "downtowners".

If I were you I'd just plant a small patch of bermuda and overseed with rye in the winter. The dogs love it and it's practically carefree. Bermuda is so easy in a hot climate. All it needs is a little fertilizer and water. If you keep it clipped really short it will keep all weeds at bay.
It is highly drought resistant and a small patch would take very little care if surrounded by gravel.