When it comes to compromise though, I think we both struggle.

Ok, so before we talk about him, let's talk about you. You said above that you both struggle with compromise. How do you struggle with it? Are you also refusing to do things for him/that he wants to do? Because if so, it's not too difficult to figure out why he's resistant to adapting this behavior: why should he do it if you're not going to do it?

If that's the case, one of you has to be the bigger person and break this cycle. Right now you two are engaging in pointless power struggles for the sake of power struggles. I'm not saying you should just let him "win" every time, but instead of "let's see my movie then yours" it might have been more productive to say something along the lines of "ok, why don't we see both movies. We can see Blade Runner this week and Lala Land next week."

If that's not the case and you're already taking that approach, then you may need to consider whether or not this guy is really a selfish asshole that's starting to show his true colors. I completely understand not wanting to see a movie that you're not interested in. But if your partner tells you it's important to them and it's not harmful to you, then those are the moments when you suck it up and you do something nice for the person you are with. You're not asking for his ATM pin number, you're just asking for him to go on a date with you. If it's more important to him that he be comfortable and not see a movie rather than do something nice for you (that requires basically no effort on his part) then I would be asking myself where else this tendency will pop up in the future. How can you build a life with someone who has the "my way or the highway" attitude?