Your device IP is only part of the data received by the DHCP server.
Your DHCP client gets:
IP address
default route (and gateway address)
distance of the default route
DNS address
BUT Netwatch DOES NOT need to know your IP address.
All it wants to know/have is a valid route to the destination (watched) IP.
This (narrow) route can be added on the fly to the route table by a DHCP script triggered by the bound condition, i.e. as soon as the DHCP client receives the data from the server, additional to the default 0.0.0.0/0 route provided by the DHCP server.
About using detect internet, it is a serious violation of Rule #5:
of Rule #5: