Worst-case scenario is that your dad does know and he's proactively asking you out to lunch to talk about it -- that's a good thing, even if it's uncomfortable in the short term.

I have tried to talk to my dad about it, and he just goes completely silent and doesn't engage and doesn't ask questions, so it gives the feeling that he's not listening, even if he is...

I would love it if my father (a public figure in a southern state) took proactive interest in my recovery. So far, not so much.

Let us know how it goes.

I'll be just fine and dandy
Lord, it's like a hard candy Christmas
I'm barely getting through tomorrow
But I won't let sorrow get me way down.