My guess would be that it's not a WAN problem per-se, but that the LAN side interface of your router is getting swamped by multicast frames flooded by your switch.
As has been mentioned in another comment, you'll need to enable IGMP snooping to enable your switch to properly constrain multicast frames. You'll also probably need to enable the IGMP snooping querier, unless you've got a multicast (PIM) router on each of your VLAN's. On a Cisco switch, you can enable IGMP snooping and the snooping querier by entering the following two commands in global config mode:
ip igmp snooping
ip igmp snooping querier
You should ensure igmp snooping is enabled on every switch in your network. The snooping querier only needs to be enabled on one switch, assuming that switch has an IP in each of your VLAN's. My understanding is it won't hurt though to enable the snooping querier on every switch. Note that for the snooping querier to work, your switch will need an IP in each of your VLAN's, or at least each VLAN that has multicast traffic you're concerned about constraining.
In case you're curious about why you need IGMP snooping:
As you probably know, normally a switch delivers traffic by consulting it's CAM table. The CAM table is populated by inspecting the source MAC address of every frame received by the switch. Every source MAC address the switch sees is added to the CAM table, along with the switch port that the frame in question entered the switch on. In this way the switch "learns" what MAC addresses are connected to each port.
The switch uses the CAM table to determine where to deliver incoming frames. If the destination MAC address is found in the CAM table, then the switch knows which port to deliver the frame to, and the frame is delivered to only that port. If the destination MAC address is NOT found in the CAM table, the frame is flooded out to every port on the switch.
With multicast traffic, the the source MAC address of the frame will be the MAC address of the multicast sender, but destination MAC address of the frame will be the MAC address of a multicast group, not the MAC address of any specific individual PC. This multicast MAC address should never normally be the source address of any frame, so in normal operation the switch will never know where to send multicast frames. It will have no choice but to flood the frames out of every port. When this happens with a really large multicast stream, these flooded frames can sometimes overwhelm other systems on the network.
IGMP is actually a layer 3 protocol designed to allow IP hosts to inform IP routers that they would like to join a multicast group. Technically IGMP has nothing to do switches and layer 2 operation, but a number of switch vendors, including Cisco, have added features to their switches that allow the switch to listen in (or snoop) on the IGMP traffic between IP hosts and multicast enabled IP routers.
Unfortunately IGMP snooping only works if there is a multicast router on the subnet(s) in question. If there's no multicast enabled router, then there's no IGMP conversation to "snoop" on. That's where the IGMP snooping querier comes in. It send out the IGMP membership queries that would normally be sent by a PIM multicast router, thereby initiating the exchange for the switch to "snoop".
It would be nice if IGMP snooping was enabled by default on most switches, but I suppose the reason it isn't is because, while IGMP is an IETF standard, there's no actual standard for IGMP snooping.