Invited on a 1st date, what would you do?

I've been chatting with a guy on Tinder, and he indirectly invited me to go bouldering. He mentioned he can't do it this weekend since he's visiting family, but suggested next week sometime. Coincidentally, my birthday falls on one of those days. I have plans for a meal with my family on the weekend, but I don't usually celebrate on the actual day. Would it be weird to have a first date on your birthday? Should I tell him beforehand that it's my birthday, but I don't have anything planned and am available, or should I just treat it like any other day? Alternatively, I could say I'm busy that day.

What do you recommend?

Invited on a 1st date, what would you do?
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