OK, Compressor should show up in the list when you click the Purchases button:
The official way to reinstall is to hold down the Option key while you click the Purchases button. I assume it looks to see if it's installed. If it isn't, then it says Install at the right instead of Installed. Again, this is assuming you've trashed the original program from the Applications folder. I can't remember if it's enough to put it in the trash, or if you have to actually empty the trash, or if it also works if you've copied the App file to an external disk and then disconnect that.
If yours is not working this way, or if Compressor is not even showing up, then I think one of two things is going on. #1 - is it possible that you have two different Apple IDs and you're signed in with the one that did not make the Compressor purchase? or #2 - there is a problem with your Apple Account and you're going to have to get help from tech support. If it's the correct account but it does not remember what you have purchased, then you need to get that straightened out.
I'm sorry that you're having these glitches, but this is the accepted technique for reinstalling - I've done it myself.