UPDATE: I thought I had perhaps found a partial way: Don't do Event invites via the Events section; instead, do it from the LP section -- which I see DOES have ONE Event email -- the custom template I created -- listed first in that pulldown. (However, none of the other standard Event email templates are in the LP's pulldown list. So my question above still stands.)
But I now think this was an anomaly, or a bug, because I cannot duplicate that. I've opened other LPs pulldown choices and those do NOT have the Event invite email at the top as a choice: