getting internet access on an a computer with two LAN connections where one doesn't have internet acccess

I was in a situation where I had two separate LANs, one for phone, one for data.  The phone LAN had no internet access.  I wanted to give the phone vendor remote access to my PC while connected to the phone LAN.

Each time I plugged my computer into the phone LAN and had the data LAN connected wirelessly, the wired connection would take preference no matter which LAN was connected first or any bridging settings I had set:

The wired connection always took priority.  I fixed this by simply giving the wired connection a static IP with the proper wired IP address but changing the default gateway on the wired connection to the default gateway on the wireless connection that had internet access.  Here's my example where 10.10.10.0 was the phone LAN without internet access and 10.0.0.0 is my data LAN with internet access:

This allowed me internet access and allowed me visibility on the wired LAN without internet access.