A lot of responses are unhelpful here tbh. What I can see from the info you have relayed is that your SIL is quite a bit younger than you and this suggests to me she hasn’t developed a lot of diplomacy for certain situations and the amount of messages she sends you I think she may be feeling isolated too. This party hasn’t been thought out too clearly. Even if invited I wouldn’t bother going.
Guys don’t often keep up contact with family the way we might. Especially if the parents are divorced there will be an odd dynamic. I think that maybe he is embarrassed by his family a bit and is trying to spare you. Albeit in a round about way.
This stage of childhood isn’t easy and lack of sleep makes it worse. It will change, this stage doesn’t last forever, I promise. Juggling work, kids and OH isn’t easy with the constant worry about money (first hand experience). Also, my mental health wasn’t good after my second (DS is 4 now and I’m (quite happily) accepting help).
You are doing well to even get out of bed some days, I know. You need to confide in someone - your mum possibly? And you need to see your doctor, you don’t have to feel like this every day especially when help is there for your mental health. Dealing with my anxiety and depression first has helped me over all areas of my life and my kids are better for it (and happier). Six months ago I had the strength to deal with my relationship with my DH, who never wanted to do anything with us (out of sheer laziness) and having tried for years with him (his relationship with his parents is terrible, although I love them).
Balancing jobs and kids isn’t easy, parenting isn’t easy and having a partner isn’t always easy but you are still together and both trying. You are both sacrificing different things and finding hard to get a break so you are obviously going to feel overwhelmed. It’s fine to have separate social lives but I would suggest you find some common ground with OH, even out walking with the kids before tea time, together.
You do need a focus or hobby to ‘escape’ to every now and again (I’ve still to find one lol).
Finally, don’t take nonsense - if you want a little time for the two of you then carve it out. Don’t let anyone treat you harshly though. But please, seek help for your depression, it will help make life better.
Ps. Sorry for the long post. 