All of Apple's "zero configuration" stuff runs over the multicast DNS, or mDNS protocol. These packets are identified as follows:
Destination MAC address (IPv4): 01:00:5e:00:00:fb
Destination MAC address (IPv6): 33:33:00:00:00:fb
Destination IPv4 address: 224.0.0.251
Destination IPv6 address: ff02::fb
But what it all comes down to is one bit: In the first byte (pair of digits) in either of those MAC addresses, notice that the least significant bit is set to 1. That is an indication to the networking gear that this is a multicast packet and should be flooded to the entire network (other than where it came in). It can get a little more complicated than that, but in general home networks are pretty simple. In most cases, everything received on one wire of a home router will be sent out every other wire (except the WAN port) and also via the wireless network(s).
It appears that the Belkin isn't doing that. I don't know anything about their support network, but you might try contacting them and let them know that multicast forwarding (in particular, multicast DNS) isn't working right.
If I login in to the router from the iMac however, I can see the ATV in the connected ethernet peripherals so it is at least seeing it in some sense, it's just not connecting properly.
That means that DHCP and unicast packets are getting through. DHCP uses a broadcast (ff:ff:ff:ff:ff:ff) MAC address, so it appears those are being forwarded correctly. None of that depends on multicast.
More detail on mDNS via this link.