Hey, that's another great idea!

In addition to all the tips summarized by Analise, how about getting the FT community to help? As Boraxo said: "How about finding a friend to fly up with the dog on a cheap weekend RT?" And as the above poster said, maybe that "friend" could be a fellow FT person!
I agree that maybe an FTer looking to do a mileage run might be willing to do it! From what I've read on this board, I'm sure that there are some people who would probably do it if it fits into their plans. Perhaps there's someone already looking at doing an MR to where you are (or close enough that you could drive to where they're going)? If there's a pet lover planning to do an MR and you can coordinate your schedules -- and of course, you're willing to pay the dog expenses (and maybe take the FTer to a lunch or get them a little thank-you gift) -- this may be a fun way to do it.
Cool... it could be similar to that serial MasterCard commercial about everyone pitching in to bring a lost dog home. I could just see it now...
Plane ticket for dog: $75
Thank-you gift for FTer on dog-run MR: $40
Having yet another great story to share with the wonderful FlyerTalk community: Priceless
Seriously, it may be worth posting a question in the Mileage Run forum and asking if anyone is planning on going that route anytime soon. Again, good luck to you, the boyfriend, the pooch, and anyone who helps!