I have to admit that I didn't think about the router acting as a DNS relay or DNS proxy.
Having said that, if it isn't working (as seems apparent) then my recommendation would still be to configure a fixed IP address and DNS servers.
What make and model is the router? The router setup should confirm if DNS Relay is in use, although some devices with no DNS Relay option will default to DNS Proxy if DHCP is enabled but no DNS servers are configured in the scope.
The status page should indicate what DNS servers addresses it has received from the ISP.
Regarding the Default Gateway address - there is no requirement to assign the first or last IP address in a range for the DG, it is simply a convention. Yes, it is peculiar for the DG to have .138 as its address, and the PC to have .1, but it is technically perfectly sound.
Perhaps it is an opportunity to overhaul the IP addressing scheme on the Juud network?
SD