I'm sorry, I've no advice on this. knock-on-wood I've never had this issue with any of my dogs.
However, I will say that any vet that pressures you to PTS a dog simply because of their age isn't a vet I would be going back to. If you don't have the option to change vets, then just know that they can only say what they (in their small infantile uninformed minds) think you should do. They can not
make you do anything. Stand your ground and don't give in. You are your babies only advocate. Tell them, "Thank you for your input, I'll take it under advisement. Now can we address blah-blah-blah, which is the reason we are here today." Stay calm, focused and give them your best "aww bless your wittle ignorant heart" look, you'll be surprised at what little defense they have against that.