If you place any other item then cobble stone on the same block as the item frame (rails tripwire hook lever torch etc.) It will plop the item off but the item frame will remain invisible, only issue is once you remove the item from the invisible item frame it will still remain invisible in that situation.

This video is made in a fresh 1.16.2 world; https://vimeo.com/454963167

Thanks for reporting this 🙂