You've injured the dog. You've offered to do the right thing and help pay vet bills. You don't need anyone else's permission or approval. Just confirm the offer and say you'll pay direct to vet. That's it.
breakIng the rules deliberately is not smart. Just get your extension. If you don't, then if stopped by police whilst on overstay it won't end well. But that would serve you right.
If embassy don't certify income, then the proof of 65k a month being deposited in your bank for 12 months is the proof you use to get an extension? Can't blame Thailand for your embassy rules.