Maybe they'll let you strap a dog crate onto the back of a mule...
Even without a dog, not sure I'd venture down below the rim in July. On every trip to the GC we've seen people stagger back up to the rim totally spent with a near-death look in their eyes. "It seemed easy on the way down..." they say.
One of the ranger talks was about hiking the CG and what to expect (30 degree temp variances between the rim and the bottom) and what not to expect (help getting out from the rangers unless they deem it life threatening in which case you'll get a $5k+ bill for the rescue...I don't disagree with the policy, but GC seems to attract people that get in way over their heads). The most popular book at the visitor centers is titled "Over the Edge: Death in the Canyon"...it's over 400 pages.
I definetly want to go below the rim, but we'll either wait for the kids to be old enough for the mule ride or we'll go rafting. On the last trip (Thanksgiving week) we almost signed up for a float trip from the Glen Canyon Dam that goes about 15 miles down river....water on that part of the river is calm enough and the raft is big enough that they allow kids as young as four on the trip. It was too cold and we didn't research kennels in the Page area soon enough...maybe next time.