Have the same issue with companions that I can't assign them to being a governor/take a party role.
From some testing I think it's not clearing the captured flag on the companion if you summon them back to your party directly using the Clan screen rather than travelling to the town they respawn in - as the kingdom I'm in is in constant war, I'd summoned three companions back to the party instead of travelling to the towns to pick them up, these three companions didn't show up for governor or party roles.
I then used the leave party member in town button in a town, left, and re-entered the town. Upon re-entering the town, I got the message about "Companion who was separated from you in battle is here in town" as you would normally get first time entering the town if you'd not seen the companion since they were captured, and upon taking them back into my party, I was then able to assign them as governor/party role.