I think it now becomes his game and he has a style of DMing that seems to include a certain amount of roleplaying in order to make CHA checks. He does have the power/right to run the game he likes, we seem to have had several threads about this.
Yes, but with rights come responsibilities. The GM also has the obligation to communicate their desired playstyle to the group before play begins, and I'm not sure that happened - if the GM had clearly explained what they expected in social interaction, it is hard to believe Doug would have aimed at being the Face.
So all NPCs react badly to Doug's character, and when the rolls come up, he makes minuses for himself by his bad style of role play. And he has no interest in talking about it...or anything...and refuses to change.
As others have noted, the GM creates the penalties, not the player. You are making that choice here.
And, you seem pretty solidly set on how you feel roleplay has to happen, and how it impacts the mechanics - you don't seem eager to change for a compromise either.
So he "wants to play a Faceman", but the only way he can do it is a completely false way.
Correction - the only way
you allow him to do it is a completely false way. It seems to be your requirements on how folks roleplay, and how that impacts mechanics, that leave him no other out. If that's not correct, here's an option for you:
Allow the player to describe the approach and desired result in a social interaction, rather than enact it personally. Just as, "I try to run the orc through with my sword," is generally an acceptable way for a character to declare a combat action, "I try to pretend I'm a minor official to get the guard to allow us to pass," should be acceptable.