Indeed, a head scratcher . . . if I get it right, this is what your setup is (clearly a representation and missing much of the interconnecting infrastructure):

and that . . . 

- the 10.136.4.xxx Device/s and the Host can communicate
- the 10.136.7.xxx Device/s and the VM can communicate
- but not between IP subnet ranges

One possibility, would be to ask your customer to issue you with a third IP in the 10.136.4.xxx range and then add another Bridged Virtual Network and present it to the VM.

Note: for the second VMware Bridged network, configure the VM to use the most appropriate subnet mask, 255.255.0.0, 255.255.255.0 or whatever. depending on the scope.

Another, possibility would be to leave your setup as is, and ask your customer to setup a network router route between the two, but I've no idea of your network topology, and as you said, as a contractor . . .