Update: I've found that it's not just minimizing a full screen VM that prevents the copy/paste from working. It's just changing the VM's window state after the initial copy but before pasting. Instead of minimizing, I would just make it normal after copying the guest file into the clipboard so that I could browse my host folders to paste into, and then I still couldn't copy/paste. I had to go back and recopy the file into the clipboard.
Still seems very flaky. Sometimes it does work.