Ah if you have fakie big flips down then fakie 360 flip should be a given, assuming you can do fakie 360 shoves (non-popped), all you need to do is pretend you're doing one of those but then at the last moment, grip the concave to form just what you're forming here (and jump).
In regular stance on certain 360 tricks like that (impossibles are another instance for me) I find that it helps to visualize that when popping what you're really trying to do is send the tail through the nose but then also back up and around, forces you to go through imprinting the full 360 and then like I was saying you can just sort of sit there over the board and comfortably catch it.
Proneness to turn the shoulders on the fakie variant also means halfcab 360 flips (or fakie bigger flips or whatever fellow kids call them these days) will come very naturally to you as soon as fakie 360 flip clicks, they're essentially the same thing whilst turning just like fakie big flip is essentially a fakie varial flip whilst turning if we can even really call it turning the act of carving or reverting back into your natural stance. Come to think of it you might not even need fakie 360 flip for those, just throwing a fakie big flip around with more force and your newly developed technique for forming 360 flips it might work. Funny how that sort of seems to take mental training, teaching your brain to accept that yes it really is a safe option if the board goes around 360 before you catch it again.
A process I could see work out for you if you have access to a simple flat bank would be learning, in this order, fakie 360 flip on flat, then take that technique to 360 flips to fakie on the bank, and then transition over to 360 flips on flat.