My DD-WRT router has lost its mind!

I just set up two DD-WRT routers, one as a WAP (working fine) and one in Client Bridge (routed) mode (the problem). Not long after setup I started seeing IP address conflicts on other machines. The event log always points the finger at my Client Bridge router's MAC address.

Neighbour table overflow

The log on my router is flooded with Neighbour table overflow errors. These start a minute or two after boot. The network is rather large, with +200 IP addresses being used in this subnet. The other router shows no such errors.

Mass ARP requests from 1.1.1.1

I'm also seeing constant ARP requests (with the problem router's MAC address) from 1.1.1.1. Seems like it's bugging everything on the network for its MAC address and then promptly forgetting it (or never receiving a response).

Configuration:

Model: Buffalo N600

Firmware: DD-WRT v24SP2-MULTI (03/21/11)

Wireless Mode: Client Bridge (routed)

I'm not sure what configuration details are relevant and I'd rather not have comments flooded, so just ping me in this chat if you want to know something.

Why is my router stealing IP addresses and how can I stop it?