Depending on how close the girls is (geographically ) to you.
1/-You do not usually need the B.Book to tax the car---I went 7 years re-taxing our car before I got around to getting a replacement book that had been destroyed.**
2- You can sell the vehicle without a B Book --usually at a lower price, I have done this twice,** last time was with a M/Bike a friend had brought from a Bar Girl and had never got around to changing ownership, left Thailand & the bike with me the for P.I. not returning. 2nd hand car dealers will take it, for the right price.
But for the price you paid for the car, if you are staying in Thailand, Just use it-- the downside if the girl resides near to you is she may try (to get someone to grab it) Upside is that any tickets etc would be her liability. Next time she contacts you tell her there are already outstanding speeding fines she, as the owner is liable for.
** Both these things I did in Issan--which is a much more relaxed place when dealing with Gov departments then BKK