I'm trying to describe a character who dances for her husband so that he doesn't get mad, apologizes profusely even if she hasn't done anything wrong. However, she is also strong-willed and determined. I considered using the word "altruist", but the character is not selfless.
People pleaser describes such a person, even the obsession part.
a self-seeking, servile flatterer; fawning parasite.
Lapdog would be one, but that has more of a political connotation.
People-pleaser gets more at the personal dynamic, but that is usually general, not specific to one particular relationship.
It seems like you are describing a dysfunctional / abusive relationship, so what the wife is doing in an immediate sense is "placating" and as a long-term psychological strategy is "fawning". There aren't really good noun forms of those; calling the wife a "placater" or "fawner" would be accurate, but awkward.
More context might be needed, but the character in question is "strong-willed and determined," though "not selfless." Most of the suggested nouns are pejorative because people have assumed that the character is weak, relative to her husband. However, she might actually be more mature and wise compared to him.
A happiness pump is defined by Simple English Wikipedia as
someone who will do anything to make other people happy even if it makes them very unhappy themselves
The term is taken from a thought experiment intended to illustrate problems with the philosophy of utilitarianism.
If the woman is constantly appeasing the husband, and the husband isn't contributing equally to the relationship or even downright abusive, then she would be codependent.
She then might still be strong willed in other regards (particularly if someone speaks badly of her husband)
This is a hard one. Calling them a doormat might be appropriate, although that usually tends to mean they do what they are told without resisting, rather than taking initiative.
I don't think there's a one-word answer.
The personality is very peculiar and may not exist as you think. I'm concerned that your character may not be believable.
Understanding the motivation (ie why people do things) especially odd or 'irrational' things is important. The behaviour/attitudes described is not one but two contradictory ones. She needs some 'reason' to give-up her inner driving role for subservience. That depends on him. ie a relationship. There are a lot of words at https://en.wikipedia.org/wiki/Transactional_analysis#Outline but don't be put off.
In short people do things because of some perceived reward. Love or triumph at being a victim, or gloating at another's discomfort. Couples often argue and both feel they've won something. For some it's the only excitement they get and plot for when they can do it again. You'll have noticed in the replies many references to abuse and so on. (That's a relationship of course.) That's what people are taking away from your thumbnail sketch. That relationship is double-ended and will change as things develop. Lots of potential so long as you can make it believable.