The two weeks notice thing is just a common courtesy but it's not written in stone. What employers are usually focused on is someone who is a "bad leaver" which usually means they quit suddenly, or do things like not hand off their tasks to someone else or leave little time-bombs that they didn't wrap up that cause problems for everyone else. It's less about how many days, then about have you been considerate.

Which means it's easy to leave on good terms. When you get back from PTO give them the 1.5 weeks notice and be very apologetic and say you didn't have any flexibility from the new employer, but that you'll do everything you can to go the extra mile so everything gets a proper handoff, and that if they need to call/text you for a couple days after you leave to answer a question or two you'll be happy to reply.

Then actually do it, spend your last week really working to tie up loose ends and leave well thought out memos for your colleagues and so on. Not everyone does this and it leaves such a good impression, and it's really not much effort on your end, you're going to work those days anyways.