I know this behaviour.
I just kept being told there was no other option and that I would pretty much have to deal with it. He wasn't going to let the dog go into care.
He also said that because the dog is unwell, there is no way he would ever consider getting him re-homed.
if I dare object to the situation, I'm just being told I don't care
This is him acting as if the only two options are the dog being got rid of, or you stopping your complaints. In other words "Shut up complaining or you are a nasty dog hater". He knows you don't want to be thought of as a bad person, and is using that to get his way by oversimplifying the issue.
I was told to stop playing victim and that my attitude towards the situation was awful? I tried to explain how I was feeling but was shut down with a very indignant tone
Unless I'm 100% supportive, he genuinely sulks or storms about the house making it very clear he isn't happy.
If I can't be supportive, then at least get off his back and say something contributory instead of doing the complete opposite of what he'd like and making the situation worse
This is him accusing you of acting like a victim, when actually that's what he is doing. He's acting like he is the victim of your nasty behaviour. As before, he knows you will try to act like a nice person, and stop complaining.
I got a text message saying he'd rather do his own thing this evening than argue.
He literally won't talk to me about the dog though; it's weird.
This is him changing the issue from the problem with his dog, to a problem with you being argumentative. Again, trying to make you feel bad about not dong what he wants. Did he also suggest that his ex wife was argumentative?
This is the consequence:
I was getting to the point where I thought I was the problem and was somehow being selfish.
He said he wanted to do his own thing so I've let him do exactly that.
I had this for many years. I'm pretty sure he honestly thought he was the good guy with a nasty, argumentative wife. He got everything he wanted and even then I often felt bad about myself, like I was a bad person. This situation is no fun for either person.
I've sice had two boyfriends and both have indicated (without knowing about any of my exh's complaints) that I am the least argumentative and difficult person they could imagine.
Watch yourself there. Talk to other people and question what's going on; question his version of events. If this is a recurring theme in relationships, perhaps speak to a counsellor.
Also give his ex wife a ring, ostensibly about the dog, and see if she has a different take on things.