Is euthanising my extremely anxious dog the right thing?

We’ve had our dog since he was 10 weeks old, he’s now 4. When he was 4 months old we noticed he was extremely afraid of pretty much everything, strangers, other dogs, loud noises, car rides etc and had very severe separation anxiety (to the point where he would physically hurt himself). We had tried socialising him and had met with trainers etc but they suggested we see a vet behaviourist as he wasn’t making any progress due to his anxiety. We ended up seeing a vet behaviourist and he was initially put on Trazadone and Clomipramine. That helped slightly but he still struggled quite a bit so the behaviourist added Clonidine. This coupled with training meant we finally got him to the point of being able to leave him for short periods of time and made progress over time albeit slow. He still struggled a lot on walks, meeting strangers and car rides were a no go despite training. We put thousands into training and still had made very little progress, he was still scared of many things and even struggled with being alone at night. Before you ask, crate training had no success and we tried very hard. Behaviourist suggested we also add gabapentin. After the addition of the fourth med, we finally got to a point where we started making progress. We were able to leave for longer periods, we could take calm walks in the neighbourhood as long as there weren’t too many people and no dogs around and we managed to have some guests over at the house finally as training started to have an impact. He is however not good around kids at all, not aggressive but would get in their face and bowl them over as he couldn’t control his excitement. We tried training, but haven’t made much progress so resorted to just keeping him separated if kids came over. I also managed to get a job where I worked from home 100% so this made him a lot happier as he didn’t need to be left for long periods. He still had bad nights here and there but rarely. We have made progress, but have done a lot to change our lives to revolve around our dog. We don’t have guests over as often, we don’t go on holidays because boarding places won’t take him and he really struggles if we leave him with anyone else. He recently started getting recurrent yeast infections which we found out was due to a grass allergy. This meant adding another med to the 4 he is already taking. The allergy meds are very expensive so it’s now costing us $500 per month just on meds. Last week, we found out that a growth on his head is cancerous. It is a mast cell tumour, and could have already spread but he needs urgent surgery. We’ve already maxed out our pet insurance with all the allergy issues. Also found out we are expecting a baby and I don’t think I’m able to deal with a high needs dog as well as a newborn and would constantly be anxious. We don’t want to send him to a shelter as with his issues he would likely spend the whole time severely anxious and then be put down anyways so I don’t want to do that to him. The vet suggested because he is such a fearful dog it’s not a good life for him anyways and we put him down. But should I be getting him the surgery and then trying to rehome him? What is the actual likelihood of him being rehomed with all his ongoing costs?

EDIT: For the people coming at me about having a baby when I can’t afford my dog. I literally have been affording my dog up to this point and have spent tens of thousands already. Just to clarify, I have the money for a baby (surprisingly they don’t cost that much when they are really little lol). I do not have the money for a baby and a dog who is going to cost a lot more for his cancer treatment plus pre-existing issues on a lower income due to maternity leave. Both his situation and mine have changed. He has cancer, I’m not just putting him down to have a baby.