This actually IS possible because the phone *can* connect to two networks at once. What it doesn't do is use both for it's Internet connection, but that's ok because that's not necessary.
1. Connect to the WiFi network you wish to use, and view the Info for said network
2. Under DHCP, note all the values that were automatically provided.
3. Switch to Static setup, and manually enter the values above. LEAVE THE ROUTER BLANK.
iOS recognizes that it cannot use the WiFi connection to reach the Internet, so it switches to mobile data. The WiFi icon will not be displayed, but you WILL still be connected to the WiFi network as well, and be able to access any resources on the local LAN (WiFi or Wired into your router's switch ports.)
This is tested and confirmed and used in a real world scenario on a 5S with iOS7, but probably works with earlier phones and versions as well. Specifically, this was used in a situation where Philips Hue light control was required while still connected to LTE. It has not been tested with AirPlay over WiFi, but theoretically it should work with that too. I was able to browse the WiFi router's admin interface while connected to LTE.
Caveats:
1. Because you've set your IP manually, there is a risk of possible IP conflict with another device on your network. This is unlikely on a network that does not have devices coming and going at all, and can be avoided if you specifically choose an IP that is outside of your router's DHCP range (If your DHCP range is x.x.x.2 - x.x.x.50, use .51
2. If you do actually want to use your WiFi connection for an Internet connection, you will have to remember to manually switch the WiFi connection back to DHCP (or manually fill in the Router, but it's easier and faster to just switch back and forth.)