If you can't narrow it down it could be a hardware problem. It is very common for wifi to gradually fail on this router. I have seen lots of reports about it in forums, and it has failed on one of the two I own. It works for a while, and then it dies.
It's very likely that the link between the two routers is the problem. This explains why those who get IP by the main router lose their IP while those who get IP by the Asus don't.
It fails gradually. It works for a while and then it fails and requires some kind of restart, or the network goes down for a few seconds and then comes back up. At least that's what I experienced.
I can confirm that the wireless part really seems to work - wlan0-1 clients get associated and get IP address from RT-N56U. I did not try if they can reach other clients on lan (like 10.11.13.2) - at the moment, this is blocked by firewall (only route to router is allowed), but I can adjust for test purposes.
However, the lan part seems to be working as well - at least for wired clients: I can connect via luci, via ssh. I think that in the past I also tried pinging from RT-N56U to the wired world - both LAN and public internet (although I cannot confirm for 100 % now).
To me it seems as if the communication is partially broken in the RT-N56 - between the the wireless and wired part. Interestingly, it is not fully broken - e.g. the router gets request for DHCP lease, but the offer is not accepted/confirmed. I also tried to restart router but that did not help. Only restarting RT-N56U does - so it indeed seems as a hw problem.
I will try to wait for another break-down (possible only in the weekend, wifi is used by family members) and do some more test to narrow down the weak point or try to dump the communication on the network directly on RT-N56U. (However, I am no expert in so it will probably not tell me much.)
Easiest two things to try would be to see if when the problem occur, does Asus have internet access? And do Asus clients connected by cable have internet access?
Just an update: I have not had much time lately, but the drop outs forced me to check again - while reading the logs from both the DHCP server and from RT-N56U I was able to spot the moment WLAN stops working.
rsyslogd[356]: May 30 16:09:12 wifi kernel: [76050.920666] ieee80211 phy0: rt2x00queue_write_tx_frame: Error - Dropping frame due to full tx queue 2