I would guess you're going to have many problems with this. First, I can't imagine it looking good (esp when you factor in color variation). Second, you will probably end up with expansion/contraction issues and may get some buckling. third, over time, I wouldn't expect the caulk to hold up and it will chip away and look terrible.
I'm sorry about your dog, but I'm thinking maybe a simpler and better solution is to try wee wee pads or something to prevent this from getting on any of the wood (eventually, it will wear down the finish). Maybe "train" the dog to just pee in one area and then buy an extra box or two to replace later (and maybe extra underlayment). Or wait until later to install here.
Debbie, The Flooring Girl
http://TheFlooringGirl.com