Yeah, I would just request the day off and go. You may be able to take the day without explanation for a "prior commitment," but I wouldn't be dodgy about it. The firm should be aware that you'll be doing OCI, and shouldn't have a problem with letting you go.
Trying to be coy will backfire when either someone from the firm or someone who knows the firm sees you there. Just be a grown up about it and things should be fine.
If they have a problem with it, they're free to make you an offer before you leave.
I agree that being coy would backfire, and that's what I meant about not being dodgy. And it's better to err on the side of seeming straightforward. I was just trying to say that it might
be possible to be completely straightforward by asking something along the lines of "Who would I need to talk to about having one day off on [date] for a prior commitment?" Maybe they would ask, maybe not. But in no circumstances would I advise being coy, cagey or embarrassed about letting people know what you're doing if you ask.