Would I be putting my dog down for the right reasons? Or wrong reasons?

I am very new to Christianity and I would like some advice (or prayer) on what to do with my dog. I have a 16 year old little dog and an infant who is about to start crawling. My main issue is that my dog poops and pees in the house 90% of the time that she has to “go.” And I cannot have this happening when my son is crawling all over my floors soon. But I can’t make a decision on euthanasia – I’ve never so much as killed a squirrel while driving, I would cry if I did. Just not sure if it would be mercy or selfishness to put my dog down.

We let the dog outside but she still goes in the house without warning and overnight. And we have very cold Winters in Wisconsin when she will refuse to step outside too. I am so fed up with cleaning up after her and just feel like my quality of life is taking a hit because of her issues. I feel like she’s just going to ruin my house that we just bought less than a year ago. She is so old that she is deaf, she can only chew soft food because many of her teeth have fallen out, she smells horrible because of decaying teeth, she limps/has arthritis, constantly licks her paws, she doesn’t play, just sleeps, etc. But the reason why I struggle with putting her down is because she still loves to eat and runs around excited about food. And I struggle because I feel like I am making a decision that God should make – I shouldn’t be deciding when she lives or dies especially when I think she still likes living just for food. I don’t know if putting her down would be “mercy” because she doesn’t seem like she’s in much pain. She has a huge appetite. But she doesn’t have interest in anything besides food. I just feel wrong about making a decision to put her down. I ask myself if I’m I putting her down for HER mercy or for ME (just being fed up/for my son)? I wanted to ask this question here to see what Christian hearts and minds would think is right (as opposed to a dog sub.)