Yep, get yourself a good therapist, one who wants to help you work towards not giving a shit.
DH says we are couple and we need to be a united front.
Just because DH says it, doesn't make it true. You are finding his mother extremely stressful to spend time with. He is not finding her that bad. This is inherently a difference and not "a united front." Side note: that phrase is generally used to mean two people standing up to somebody together, not in the way he is saying it, which is "please shut up and be a good little stepford wife". I find his attitude quite horrendous actually.
And if I'm not there he would need to tell them why I'm not coming to stuff.
So what? "DW can't come today because <benign excuse you have agreed beforehand>." Job done, no drama needed.
So I feel like I'm "locked" in.
You feel like you're locked in. The reality is you are free to do as you like. Take this to your therapist :)
He makes me go to other events with his friends as well, which I don't like.
Take this to your therapist too. Your H can't make you do anything, unless he's physically manhandling or threatening physical harm to you if you don't. If that's the case, you've got even bigger issues that you also need to talk to your therapist about.
But all his friends are quite public school in a caricature sort of way, and I don't find we have a lot of common ground.
I really do think it sounds like your H is from a background where women are not people in their own right but trophies to decorate his arm with to fit in with what mummy and daddy and his public school mates expect. I'd hate spending time with them too. The good news is that you don't have to. The bad news is that you appear to have stumbled upon the news that your H doesn't see you as an equal who has her own autonomy.
He says he hates going to stuff alone.
And you hate going to stuff that involves his mother or his mates. But he expects you to shut up and play along doesn't he? Because he sees his desire to not be uncomfortable as more important than your desire to not be uncomfortable. That's not equal, and that's not fair.
There is a middle ground compromise to be found, but you're going to need to do the work (in therapy or otherwise) to a) let MIL and public school boys wash over you, and b) learn how to stand up for your rights and what you want in your marriage, and c) recognise and be okay with a period of things being a bit uncomfortable as you start asserting yourself and your H/MIL/all sorts of other people will push back to try to keep you in the neat little box you have been conveniently sitting in. It takes tenacity, strong will and a willingness to let other people and yourself be uncomfortable to change status quo dynamics, but the rewards are there to be had for those who put the effort in.
Incidently, you also mention being very busy in a high stress job. Your overall stress burden will be contributing to how much your MIL stresses you out. If you prioritise relaxation and rest as a non-negotiable essential part of your life, everything is easier to deal with, I promise.