The "wannabe-posh" personasounds to me like a wall he is putting up, because behind that wall he is struggling and vulnerable. My DH has a "wannabe-posh" front that he puts on under pressure, as well - I know when he is doing it, but others don't. He does it because he learned it when he was younger and under stress, as a way of appearing to be coping when he had suffered a bereavement and wasn't offered any support. It "worked" then, so he uses it now when he feels like he isn't coping as well as he should be.
I have a "sarky, jokey cynic" persona that functions in much the same way. DH knows that when I am behaving like that, I am starting to crack and am throwing up a wall to maintain normality, for the children, for work, for other family etc. I think you need, in the longer term, to work with DH so that the two of you can agree that yes, you are sensitive to these changes in his behaviour, that you want to support and reinforce him, not expose him, and that he can let you behind that wall and let you be there for him.
First, though, I suggest that you get some support for yourself. You ay not be the anxious type - but you don't have to be to suffer anxiety, and having some anxiety now would be a normal reaction to the stress of coping with a spouse who has MH issues and who isn't allowing you to come into his bubble and support him. It doesn't mean you'll have anxiety forever. I'd have a chat to your GP, if you have one you can talk to?