DHCP broadcasts are only visible on the same network segment.
A DHCP relay forwards DHCP to another network segment, regardless whether that would use a separate subnet or not.
If your second AP would constitute such a segment, a client's broadcast for a DHCP server would never reach Pi-hole without a DHCP relay.
But that may indeed not apply here, since you've clarified that a client may succesfully acquire a lease through either of your APs (not just one, as I read your initial statement to mean).
I don't even think it's DHCP nor DNS then, it woud be wireless routing.
If your network is indeed just one segment, then a roaming client already holding a valid DHCP lease for the same network/SSID wouldn't be strictly required to acquire a new lease when switching from one AP to another.
All that has to be done is updating routing tables. Your Wifi APs along with your router and your client should take care of the hand-over.
Do you have any hints that your client would try to (re)negotiate a DHCP lease at all?