Ok, so Positron himself recently confirmed that it is only possible to ever have one patron badge.
"There really is only "One" patron badge, it just has different flavors. So if you swap patron pools, your badge will change as well. No one in the game will ever have more than one patron badge, because that simply can not happen. "
It's always been a minor annoyance to me that this site thinks I am "missing" the other three. Especially the low hanging fruit. lol...is there any way you guys will change that?