I have the same problem. I need to add another group to the list. I've selected the group to allow Zoom, but on initial entry that group is not one I can select. I need to get it on the original list.
I have figured out a way to make it work for me (eventually!).
Log into Zoom
Prepare to go live
When you get to the page where you select the FB group etc, right at the bottom, underneath the 'Go Live' button, there is some tiny writing.
Click on 'Facebook Security Settings'
Continue
Decide who gets to have access, then click continue
Click on 'Choose what you allow' and you get to a list of all the groups you are in. Tick the ones you want - job done.
Hope it works for you! 🤞