I've only read your posts @Poptart22 - but from your responses and quotes I think you've done a great job of sticking up for yourself.
The answer on threads that involve difficulties and pets is always 'get rid of the pets' as if they are some kind of disposable commodity and not much loved family members!
Posters who have said it will get easier in a couple of years are right and there are things that can be done to ease the burden in the meantime, but given how much anger and resentment this is causing you, the real question is, do you still love him and want to stay with him?
If he got a new job so he was home 7 days a week, would everything be OK?
You say he took the promotion without any real consultation with you so is the anger and resentment really about the stress when he isn't home or is there something else going on?
Lying about travel to take a promotion which you wouldn't have wanted him to if he'd been honest is symptomatic about other issues.
If it had been something you had decided together and gone into with your eyes open the busyness and stress would still be there, but the anger and resentment wouldn't. Even if it is worse than anticipated, in a healthy relationship, he would want to help find a solution (not necessarily getting a new job, but at least exploring options).
If you do love him, want to stay with him and the stress of three nights alone is the only issue, then I wouldn't necessarily be pushing him to look for a new job (because that will only lead to resentment on his part). He may be refusing to discuss it because he doesn't want to keep being told he needs to find a new job.
Ask him to sit down and talk about options, with the promise that you won't ask him to look for a new job.
Look at finances together to see if you can get more dog walking time. Could you move the use of the dog walker to the evenings and find time to walk them yourself in the mornings?
On the nights when he is home, can you get an evening 'off' where he takes care of everything (putting your son to bed, dog walking etc) so you get the same total downtime that he does?
Can he change his evenings away, so they don't clash with your work evenings, or vice versa?
It sounds like you have both hit an impasse - if you want a find a way through it you are going to have to find a compromise that isn't as extreme as getting rid of pets, new jobs or divorce!