You'll need a DHCP server on both sides of the bridge to implement redundancy properly. XP Clients are not servers.
I suppose there's possibly some software out there that will do it to some degree but it probably won't integrate with AD correctly and so-forth (and what happens when the "nominated" workstation is broken or just switched off?).
How long do you expect the bridge to be offline? It might be simpler to extend the DHCP lease length a bit, and IIRC clients can just continue to use their currently allocated address if they can't contact the DHCP server.
If you want to implement DHCP redudancy with 2 DHCP servers, its easy enough. Just think of a scope thats potentially twice the size you need it to be for all workstations, and allocate half each of it to the two servers so there is no duplication of address ranges.
Clients will get IP addresses from whatever DHCP server responds fastest to a request normally, and with the bridge down this will simply just be the server on their side of the bridge.
In the remote office, you could get a low power / low cost device to act as your "2nd" DHCP server and shove that into whatever cupboard or cabinet you currently have the bridge hiding in. It doesn't have to be a full price server with all the trimmings just to do this.