If something is painful you aren't doing it often enough, right? So my (completely uninformed) idea would be 27 bit addresses that are only routable on the local LAN and then a second optional 27 bit address to route between LANs on the WAN. The effective 54 bit address space would have been more than large enough, and if you support modularly extending addresses like that then there's no reason not to keep going beyond the initial 2 by eating into the payload.

Being completely uninformed I have no idea how severe the negative consequences of this scheme would be for the efficiency of routing hardware but I assume it would probably be catastrophic for some reason or another.

That's very loosely how IPv6 works. Your ISP will typically assign your router a prefix and will route any address starting with that 56 or 64 bit prefix to you. Then devices on your network pick the remaining bits and they get their full address.