@Bow hmm... ever heard of the 'sunk cost fallacy'? It's "the phenomenon whereby a person is reluctant to abandon a strategy or course of action because they have invested heavily in it, even when it is clear that abandonment would be more beneficial."
As an example, it's like let's say you bought concert tickets. On the day of the concert you're sick as a dog. The sunk cost fallacy is believing you should go anyway despite feeling like crap because otherwise it's a waste of money. Thing is, the money is spent either way, it's not coming back, and it's entirely likely that going to a concert when sick is going to be absolutely zero fun.
In your situation.... I know it might sound a bit callous when we're talking about an actual life, but basically I'm sorta saying.... just because you spent so much money at the vet, doesn't mean you HAVE to persist; the notion that if you were to put him to sleep now, it would mean all that money was 'wasted' is the fallacy, because... well imagine how you n your mum would feel if you hadn't at least tried? As in, don't let what has already happened dictate what happens next. Focus the convo on what is best for balancing your (as in yours, your mum's and your D's) needs with his. Does that kinda make sense?
