I’m disappointed that things haven’t worked out, however I won’t be bullied by my wife into just doing things her way. So now neither grandmother is coming.
In other news I’m rather tired of my wife “feeling prisoner in her own home” when I teach piano lessons, because she has to stay in the front room with the kids and dog for a couple of hours. Each time I say, “I’ll start going to them at home instead” (would be a relief to get out the house tbh). But then she’s like, “can we STOP TALKING ABOUT WORK! It’s all you ever talk about!”
Good that you won't be bullied, but now, your parents still don't get to see the children.
Your wife might resent accommodating your job, but she benefits from your income.
I understand the tendency to just give in to these "requirements" as it's mentally exhausting to constantly try to maintain a boundary. In addition, you are also focused on the children at the moment- as they need constant attention and taking on an argument is sometimes just too much.
Even with gasoline costs, it might be good to get out of the house for a bit, if the children are OK but I wonder if your wife can cope with them on her own. While her self image is that of a competent mother, the ages of your kids are such that they need constant supervision and it's normal for anyone to feel overwhelmed at times with small children. However, your wife may not be able to admit to that and so makes up some other "reason" for why you can't leave the house.
Sometimes my BPD mother will make a request that makes no logical sense. This is because the request is not really about that but some other need that she doesn't want to admit to. There can be a "hidden agenda" to her requests and so when I reply with a logical response, she may get agitated at that and it doesn't make sense to me until I remember the
request may not really be what she wants. Not that I can know what she really wants. None of us can read minds. Eventually though, what she wants becomes apparent as the conversation is complicated and with drama and I figure it out, or in time, I see what she wants. The clue for me that there's another agenda is that, the conversation doesn't make sense and logical replies result in her being agitated.
For instance, the issue with your mother isn't about your mother. It's your wife feeling jealous that her mother isn't as attentive and also jealous of your connection with your mother. So not having your mother visit takes care of her jealousy but results in your mother not being able to come. The issue over needing to stay in the front room during your lessons may be more about feeling jealous of your time with your students, or also her feeling inconvenienced.
It's understandable she doesn't want you to leave the house because she'd be alone with the small children- but the reality is that, someone needs to earn an income to meet your family expenses. I also wish my H could have been home more when our kids were little but he needed to go to work, and needed to be able to do that. All parents need to work out a child care/work schedule.
It's hard to know when to give in to keep the peace and when to hold the boundary. Saying "no" to my BPD mother or holding her to any factual accommodation is very hard as she's driven by her emotional needs. However, there are physical needs, such as shelter, food, heat, and bills need to be paid to attain these needs and to meet these, someone needs to have an income. So your wife needs to decide to either stay in the front room or stay home with the kids on her own while you go out.