Even if everyone is wrong and it’s not just sibling jealousy, and tricky 4-year-old age-appropriate behaviour, a diagnosis of something will still only mean your own parenting behaviour and strategies need to change to manage things better. The diagnosis won’t fix anything, but the behaviour management strategies will
This, I am afraid. I have brought up one child with MH issues which often translated into tricky behaviour. It took a long time to get her diagnosed, but even then the diagnosis didn't result in a short medical treatment that made the problem go away. The work still had to be done on a daily basis, not by medical professionals but by her parents. I also needed counselling on the way. It was helpful, but at the end of the day I still had to do the work.
You've had lots of helpful advice on this thread, OP. If you structure it up, you might end up with a checklist like this:
build a more positive general relationship (lots of brilliant advice up-thread)
set her up to succeed- by giving her little things she can do and praising her when she does them right
see if you can avoid triggers- not by letting her do exactly as she wants, but by preparing her and distracting her and making it easier for her to do the right thing without conflict
accept that sometimes conflict will happen and that is not a failing on the part of either of you- have a plan for how you will handle it when it does
think through different scenarios in advance- how bad is x behaviour and what kind of punishment/what level of anger does it deserve? don't wing it- chances are, your reaction will end up being more about you than about her
distract yourself when you feel your anger is building up (I used to put on the kettle or out the radio on or even burst into song)
get your husband to understand that he must take a more active part in parenting and that you must sit down together and plan this
try to find something in your life that isn't about the children, something that makes you happy or engages your interest, something that you can use as your own mental escape place