Raspberry pi cannot ping router or internet addresses over wifi bridge

+1 for the detailed problem description.

As I suggested on the thread you opened in raspberry pi , you could check if your main router is listed in the RPi's arp table : arp -n or if you have the iproute2 installed: ip neigh.

If needed you can add the router in the arp cache with this command : arp -s <ROUTER_IP> <ROUTER_MAC> and see if you still have the problem

You can also check if your RPi sends the ARP request as expected by sniffing all the ARP packets. On your RPi, run : tcpdump arp

You could also run the same command on the DD-WRT repeater and on any other host connected on router 1. As the ARP requests are broadcasted you should see them across your lan.

I had the same issue when installing new Wifi Repeater. The compromised solution is set the static IP for Raspberry Pi.