I don't believe I have ever seen a DHCP client that supported that option. But your point remains that it could work. In fact, the bridge could do some packet modification to DHCP packets to help its clients use it transparently. I don't think we'll see that happen, though.
In theory, it wouldn't have to modify the packets. If it was a "true bridge" and not just an AP/client relationship, the APs could figure out which MAC addresses live on which side of the bridge, and tunnel whole Ethernet frames across the wireless -- for complete transparency even to "dumb" DHCP clients and servers. Again, I've no idea whether this actually happens in practice, but OTOH presumably there's
some difference between AP/client and bridge modes, as all those screenshots have them listed separately...?
Peter