I can see the green FPS numbers, yes. I can also still toggle them visible/invisible but when I try to record it just does nothing, no error, no prompt, no response. I tried a couple other Origin games and didn't have an issue recording, even with origin.exe open. By "close origin.exe" do you mean close the window, or exit the application? Closing the window doesn't change my result, and if I exit the program Apex Legends prompts me that Origin has closed and exits the game.
Edit: *side note* when Bandicam detects Apex Legends, next to the magnifying glass where it normally says "Please click on the recording target window." Instead of having a program name there it has "() - 0x0" without the quotes.