I do OP because I've been in precisely the same situation you have. Just after my DS was born, I caught him wanking as I lay exhausted on the sofa after breastfeeding late one night.
Like you I had every single negative word running around my head. Plus exhausted. Drained. Little money. Hurt. Porn by now had become a running 18 month cycle going on for ten years. Now I had two little ones.
A voice in my head said, 'Leave it for now, pick your battles'. And I did. When I was back on my feet, in a professional established job, I went.
I remember once PatriNarc saying to me, a relationship can survive a fuck up. Possibly two. It's when it becomes cyclic.
In my 12 years on the relationship boards, I've seen a sea change in how women view porn in the context of a relationship. I used to hate it, now I do not care because I'm getting affection and care back from him, do you see?
Fubdemebtally, the usage of porn in a relationship where one partner, or both!, is feeling unattractive is unhealthy. Again, I LNOW how it made me feel when he wanked himself silly then couldnt get hard. I was really quite overweight and felt as sexy as a pile of blamonge.
It's clear he never stopped viewing and has lied. But is this a deal breaker? Only you can decide but you havnt delineated any other behaviours except touching on that it appears you ate doing pretty much all the childcare. Now that's problematic.