After calming down, and after listening to everyone's sage advice, I realized this was a pretty straightforward, if nerve-wracking fix. I un-grafted the two pieces carefully. My original plan was to put one side directly back on a needle -- until I realized I was saving the side with the hole. So I just picked out the grafting yarn.
Once I had the pieces separated again, I set about identifying a good fallback position. The 11th row of the pattern has a distinctive 3-stitch decrease bordered by several groups of 5-stitch stockinette sections. For each side, I marked one of those 3-stitch decreases, frogged back to the row before, and then slowly removed Row 12 as I carefully put Row 11 back on the needles. I had a few minor mishaps, but was able to fix them.
So now, I have two pieces, one with 16 of the 17 pattern repeats intact, and one with 13 -- the side with the hole. Turns out it must have been a slipped stitch since the yarn wasn't broken. The dogs are no longer canines of interest in this case.
Shouldn't take too long to catch back up. Thanks, everyone, for the advice, encouragement, and sympathy. Onward.