When there is an issue in an otherwise healthy relationship in my view it is best to stop for a moment and try to be a bit critical of oneself and forgiving of one's partner.

With that in mind I propose to look a bit critically at what you said.

It was jarring to me to hear that in essence you offered her money to clean for you. This approach threatens to turn her from your partner into your maid.

She has four days off but the fact is that she is working a full time job, just like you are. I imagine that working 12 hours is stressful, particularly if she works in a hospital or other setting where critical mistakes can readily be made. I am not hearing you give her equal respect as a full-time worker whose time off is just as valuable as your own.

If you told us that you came home and drank to relax after work for a few hours (how long is a few hours?) many might wonder whether you are misusing alcohol. Is it possible that you are misusing video games to address your work related stress?

Lastly, having a forgiving attitude to one's partner is quite different from your claim that she wants to argue about this all the time. Rather than speculate in a negative manner about her motivation, what if you assume that she is doing the best that she can, just like you are? And act on that assumption.

I do question some of her behaviors as you report them. She is passive aggressive to do the cleaning and then "throw a fit" (a disparaging term) about it later. Your comment about how you are not allowed to play video games when you are together I think is worth looking at. It sounds like you carry a deep resentment about however it is decided that you cannot play on the two days you are together until she is asleep.

She might carry resentment as well about the way you speak about money. When you say that you never mention it "...until she throws a fit" (again a disparaging term) about cleaning acknowledges that you do bring it up.

I would like to think that these differences could be amicably resolved with the good faith assumption on both sides that each is doing the best that he/she can. If not, I wonder if you would consider couples counseling.