Itch support just got back to me.
They say to create a Coupon for the second game, with a Membership Requirement set to the first game. This will then show automatically on the second game for anyone who has bought the first game, but not for anyone else.
For reference, I created it using a sale (rather than a coupon) and it looks like it works in basically the same way, right down to a specific URL for the sale.
There doesn't seem to be a way to limit it to people who bought the first game before the release of the second.