Surely you need the ground. I think the "C" at the top of the terminal strip is the common or ground terminal. In the owner's manual the wiring diagram, "Optional Common Lead" shows a wire from the "System Common #1" and "System Common" being tied together on the left (in the diagram) to the "Chassis Ground" (that is the symbol for ground in the diagram) On the right (in the diagram) a dashed line shows a wire (probably the green one you have not connected?) optionally connected to the chassis ground. Being extra careful, I'd check continuity between the green and the negative on a twelve volt place in the fuse panel.

I've installed the Honeywell https://www.loveyourrv.com/rv-digital-thermostat-upgrade-mod-explained/
But, here is a PDF that a person shared with me on installing the LUX - https://www.loveyourrv.com/wp-content/uploads/2013/11/LUX-DMH110a-wiring-schematic-UPDATE.pdf

Thanks for the reply, So I think you are saying I need to run a ground wire from the chassis to connect with the green wire. I notice in some Lux models there is a C pin for the common but this one was left blank.

Thanks ray, I studied the video but PDF but couldn't find where to run the ground wire. And on his Lux he had a C pin for the common.

Not sure about your setup but with my Suburban SF30 furnace and the Coleman Mach 3 AC I didn't need any ground wire. All the digital thermostat does is apply the 12v RV power to either the furnace or the AC wires via a relay inside the digital thermostat. The onboard alkaline batteries supply the relay power.
But you have a Dometic so maybe something is different.

I'd guess the green wire is a ground. That's why I suggested testing continuity between the negative (12 volt fuses) and the green wire. Got good conductivity or low resistance? Then it may be a ground. And you're gonna hook 'er into the "C" terminal.

Thanks for the response, I reached out to Lux and they said that this unit was designed for an AC system and not for a DC RV so they said that it won't work for what I want so I'll have to look for a different Thermostat.

Check this out. Looks like for the Dometic the green goes to RC-RH
http://www.modmyrv.com/2012/01/03/mod-100-honeywell-digital-thermostat

"My wiring was as follows: yellow wire (always A/C) to Y. White wire to W (for the furnace). Basically all furnace wiring is power wire to white, touch your power wire to the white and presto the furnace starts and cycle thru for about a minute (hint- twist them together to keep you and your loved ones warm while you figure out what else to do) (another hint- twist the yellow and your power wire together to keep you cool headed instead of the furnace).

If you twisted the red wire (typical power) to the yellow or white wire and it did NOT start the A/C or furnace, then try either of them on all other wires until it does. THAT wire is the powered wire and will go on to the R or RC terminal. My case it was the green wire to R/RC with jumper in (u shaped wire between R and RC [furnace and cooling circuits]) Now both A/C and furnace have power when the switch on the bottom is on heat(R/RH) or cooling (RC). Brown wire to G (which runs the A/C fan)."

Interesting article, I'll test it out. Thanks Ray

I would check the manual for the new thermostat. Some of the new digital have to be setup internally for the type of system your running. The other thing is cross check the terminals for colours. Sometimes the colours that are use used are wired to no standard. I found this in changing thermostats in residences that the color code depends on who installed and the brand of furnace.

I have a very small trailer & use a portable room AC unit & upgraded to -this- but it's just for the furnace. Huge improvement over the crappy OEM style with the spring in the back. Easy to install also - 3 wires I think.

I asked Lux Tech Service and they told me that the model I had was for AC power and would not work in an RV with DC power so I'll look for another thermostat. Thanks for all of the help.