Like a girl who has been rejected for half a year. Today is her birthday. I want to send her a happy birthday message, ok?

I think this is ok. After all, you confessed six months ago. It's been a long time. It's his birthday. I think it is entirely possible for you to send her birthday wishes. You can also confess again on her birthday. Try to see if she regrets not agreeing with your confession. Everything needs to be tried. You never know what the other person thinks unless you try, so I think you should be bold and send it to her.