First of all, make sure the proxy configuration (on the device) doesn't have exceptions for LAN addresses (192.168.*.*, 10.*.*.*, etc.). Lots of times, LAN addresses are by default excluded from proxying, because it's assumed you only need (or want) to proxy Internet traffic.
There are a few ways, but one I've used before (which doesn't even require rooting the device) is to use your PC as a WiFi access point, and set up forwarding rules on the network interface (look up "iptables forward", or it's probably possible somehow on other platforms) so that any traffic originating from the device is routed to the proxy (which you run in invisible mode, listening on the port(s) that the app tries to connect to or at least that you forward those ports to).
In this case, since you need to let the device communicate with the router (and therefore it probably doesn't work to make yourself an intermediate router, which the above approach usually does), you could instead ARP spoof the device to think your PC's MAC address is the MAC address of the router. Packets addressed from the device to the router will instead be routed to you. You can then set up Burp (in invisible mode) listening on the ports that the app connects to, and relaying the traffic to (and from) the router after interception.
A third option, which probably doesn't work here in particular but does for some other cases where not all traffic is getting proxied, is to use DNS spoofing. Obviously this only works if the device is trying to connect to a host by name, rather than by IP, but it's generally very reliable because you can set a system-wide DNS server when you connect the device to the network, and make that DNS server be your PC. Then on the PC, run a DNS service that allows custom responses (and presumably forwards everything else to the PC's normal DNS resolver, probably an upstream server), and point the names you want to proxy at your PC's address (with Burp running, in invisible mode, listening on the relevant ports). You could also do this by poisoning the DNS cache of an upstream server (such as a router), but why bother when you can directly control the DNS responses the device sees instead? Obviously that's not reliable if the software has its own DNS resolver that doesn't use the system DNS configuration, but that's much rarer than software that doesn't use the system proxy configuration.