You do not have crippling social anxiety if you have managed to live with one person for two years, get yourself pregnant, etc. If it is not crippling, in general parlance, or, does not cause a significant functional impairment - the same thing in psych jargon, why would you tell him, and especially now when your life is about to change so much?
Plus, if you tell him now, he would likely write if off as hormonal or related to the anxiety over the upcoming birth experience. He may even be right in that

.
Do you have names for the baby yet? Are you ready for the birth? Those would seem more relevant aspects of your life now - put the idea of disclosing whatever you call social anxiety on the back burner.