For me, happiness is very much like the feeling you get when you've climbed a mountain peak in bad weather: you're maybe exhausted, you're probably wet and scratched and aching, but you got through, you made it, you set out to do something difficult and you did it.
I fell in love as a teenager with a man I met on a foreign holiday, we had no money, no future, but eventually after 10 years we were able to get married and, despite money worries and other worries, it was a happy marriage. That is happiness, looking back thinking "I fought for that and it was worth fighting for".
We had a child who turned out to be disabled and with MH issues: I gave up my career to care for her and she is now an independent adult, studying the thing she cares for most in the world. I fought for that and it was worth fighting for.
I started a big project a year or so before I realised she was disabled: 20 years later and I have just finished it. I fought for that and it was worth fighting for.
I am restarting my career now, in my mid-fifties, prospects are looking pretty bleak but I know it's worth fighting for.
But that definition of happiness also means it doesn't exclude pain. I can experience high anxiety (I do) and still have a sense of happiness. Because I know I am pushing myself through the brambles up the hillside; I may not reach it but I am attempting the climb.
I am very much of the "life is a valley of tears" school of thinking. But I don't necessarily feel that excludes happiness.