So the iPad will connect to the strongest signal it can find from the network. If you have 3 routers on the same network, and you can see the air AirPort Express when you're close to it and AirPlay to it, but the same doesn't happen when you're closer to a different router, then it sounds like the middle router (or first) is blocking ports. Check the 1st and 2nd router and unblock ports 80, 443, 554, 3689, and 5353 per Well known TCP and UDP ports used by Apple software product
I have checked the configuration of the two routers. I can't see any settings that would block ports (in fact, I can't find any settings that even mention ports). Most security settings (apart from WPA2 Personal) are unchecked. I tried disabling the SPI firewall to no avail.
Do you have guidance about which settings could be used block ports?
Do you think it would be worthwhile to see if the problem goes away if I disable all security for the routers?