Edit: ****I just read your post again and I see you have power at your red wire. If this is true then you should be able to disconnect the red and white wires from the thermostat. Then in your hands just touch them together and the furnace should fire. If so then I would suspect the thermostat. ****
Leaving original text below as it might help someone else in the future.
The wire that goes to the R terminal (sometimes they use the RW terminal) should have 12 volts (battery voltage) on it. I'm assuming in your trailer that's the red wire you are referring to. The W terminal goes to the heater. (I'm assuming that's a white wire.) All the thermostat does in heat mode is connect the two wires together just like you did by joining them with your hands. So if you have no 12 volts on the red wire then the furnace won't fire no matter what thermostat you have. Here's a couple more tests you can try and they're free too. :)
First thing when checking voltage is you have to know what you are checking for. Take your meter to the battery and measure the voltage across the positive and negative terminals. This will give you battery voltage. Some people might tell you that all you need is 12 volts and you are good in your checking. But what if your battery reads 12.76 volts when you check it, and your heater reads 12.2 volts? That would be a sign of something wrong. But if you have no idea what voltage the battery has then 12.2 might look good as it's 12 volts.
Since you have no power at the thermostat I would suspect a bad fuse. look at the fuses in the fuse box. A careful inspection will reveal that there are two holes on the back of each fuse, one on each side of the amperage rating. (silver dots on either side of the 10 in the picture below).

These holes allow you to probe the fuse for voltage without disconnecting it from the holder. Check for voltage from each hole in the fuse to a good ground or the negative post of the battery. Anything less than battery voltage on either side is bad (either fuse or holder).
If you have battery voltage on both sides of the fuse you're going to have to find out why you are not getting battery voltage at the thermostat.
You could also use a long wire and grab 12 volts off something else like a ceiling light and run it to the W wire at the thermostat. If the furnace fires then connect this temporary wire to the R (Or RW) terminal of your thermostat and check the furnace again.
I think you thermostat is fine and you somehow lost the 12 volt power to the furnace that is needed to run the heater.