Hi is the xh9005 the same as x9005f?What you're asking for isn't possible within your budget I'm afraid. I'd consider looking for a refurb Sony XH9505 or XH9005 from their refurb store.
See: Bargain - Sony centre direct refurbished store thread
The XH9005 is a mid tier TV, so falls short with HDR being too dim, it does however support a 120hz input.
The XH9505 is a high tier TV, which is more suitable for HDR usage, but only does 1080p 120hz, or 1440p 120hz if you have a PC.
The refresh rate of a TV, the refresh rate the TV can accept and the fps sent are 3 different things. Many games are still 60fps, therefore running 120hz isn't that beneficial.