IMO: in a way, what he was dealing with was even harder.
If you're ashamed of something you have done, you can do otherwise in the future. If someone has been hurt by something you have done, you can atone for your actions... you can make amends. If your behavior has offended, you can demonstrate that you can behave differently.
What Al was struggling with here wasn't what he had done, really. It was what he is (or, at least, what he perceived himself to be, from the time of his birth and childhood). That's not something that one can change (at least not in the literal sense, and lacking a time machine).
(Remember also that Al is English, not American... these are related cultures but not identical. As I understand it, class-consciousness is somewhat more strongly (or at least more overtly) rooted in the English psyche than is the case here in the U.S.
Horatio Alger was an American, as was his "rags-to-riches" story pattern with its message that poor beginnings were no barrier to success.)
So, Al had gotten into the habit of concealing his background from everyone, through the decades... he'd created a new, adult persona through which he could move through society and be accepted. I imagine that over time, the habit of concealing his origins just increased the tendency to think of them as That Which Must Never Be Revealed. The more time went by, the more people who got close to the "new Al", the more he stewed about it, and the more catastrophic a "coming out" must have looked.
It took the stress of his guilt about lying-by-omission to Daisy to crack him loose... and it took her issuing several swift kicks in the tail to get the message across that his family origins truly aren't a negative thing to her.