"The rest of the department is baffled as, aside from me, there are several other much more qualified associates. He, however, although lacking in skill is adept at kissing a**, something I have neither the time nor the inclination do so. "
Rosie, you say that there are other people who didn't get the promotion too and are qualified. Are you sure that this is really about your flexible working and not the fact that this other guy is great at sucking up? As Xenia said, awful though it is, opting out of that is not an option if you want to progress. You simply can't say "I've got better things to do". That is HOW you make partner in law firms.
I know that HR have told you that the flexible working is a problem, but are you sure that they are not (in their blunt, insensitive and potentially litigation inducing way) saying "you need to network. You need to build emotional capital with the partnership. You need to be visible." Put through the idiot translator of many law firms, that comes out as 'you shouldn't be working part time or from home'. Getting to a senior position in a law firm isn't about just doing the job well, unfortunately. It's a partnership, and that's a very different beast to a limited company. It's about making the other partners like you and want you in their 'gang'. Yes, maybe that is wrong. But it's the way it is and the only way to change it is if, over time, the people in the 'gang' become more diverse. Suing just gets a pay off (I know you've said that you wouldn't do it, but along with moving on to different career options, it is basically the other choice).
Is there also a bit of an issue with you being married to a partner? I know that a lot of firms are very sensitive about having spouses in the same firm. In many, one or other has to leave. You are working full time now. Have you thought about going somewhere else and then asking about flexible working once you're settled in. Full time with one day a week from home isn't actually a big ask, particularly since I assume you have childcare every day and could therefore be flexible about the days if there were hearings. A lot of firms would LOVE a flexible working request that simple - rather than "I want to work 3 days only"
I am sorry if this sounds mean. It's not meant to. I'm just trying to add a different perspective to what others have said.