Wow, that's ambitious. You could DO it, but I highly doubt you'd enjoy a trip of that many miles in three weeks. I think you really need to choose either the Grand Canyon or Yellowstone and create your loop from there.

Given the cost of gas at this point, you might also consider flying west, and renting a car out there. The dogs might present a problem with that scenario, though. Not sure what the airlines are charging for pets these days, or if you're willing to let them fly in the cargo hold area. If it's an option you'd consider, I'd look into the specifics. This August, we're flying to Vegas and driving from there. It worked out to be a pretty significant cost savings and gave us more time at our destinations. (No dogs on this trip for us, though. We're hiring a house sitter to stay with them for a few weeks.)

Good luck and have fun with the planning...