Jumping ahead on the steps of attempting to diagnose without furnace removal.
My suggestion is to remove the furnace.
Turn off the propane tank(s).
Run a range burner until the flame goes out.
This will prevent dumping unburned propane into your living space.
Pull the 12 volt fuse dedicated to the furnace.
In most cases, you can remove the furnace by removing the grill on the return air intake. There is also likely to be a removable cabinet panel (4 screws) to give you "easy" access to the furnace cabinet. That panel should be large enough to allow the furnace to slide out of the cabinet.
Typically, one must disconnect the propane line from the base of the furnace. Next there will likely be several flex ducts attached to the furnace body...in most cases, these are removed by simply rotating them counter clockwise about 1/4 turn. Then there will be several screws holding the furnace cabinet to the floor. At the rear of the furnace there is typically a bracket mounted on the floor and a lip on the bottom of the furnace body slips under that bracket to secure that side of the furnace body...no screws. Obviously, every installation is different, but just remember that all this stuff is designed to be assembled and serviced easily. Take your time and proceed cautiously, but it should be pretty easy.
In my experience, your furnace exhibits symptoms of a clogged/dirty or failed sail switch.
VERY IMPORTANT CAVEAT. The sail switch operates by the circulation fan building enough pressure to move the "sail" on the sail switch enough to close the sail switch. I had a situation where the switch itself had NOT failed, but the switch resisted operation by the sail despite being clean and lubricated. The fan would come on, and the sail was free to move, but it would not depress the switch enough to close the circuit. I tested my sail switch repeatedly, and when operated by hand, the switch functioned, but it did NOT function when being operated by the air pressure and sail.
My point is, get the sail switch part number and order a new one (or buy at an RV store) regardless of what you find or think you find. It's enough work to pull the furnace to warrant thowing a new sail switch at it to avoid the 3 times I had to remove my furnace because of the nature of the problem. I fiddle/farted around with mine twice, confirming that the switch worked by using the ohms test with a multimeter. The switch worked when I had it in my hand, but the sail could not operate the switch...a flaw in the sail mechanism of the sail switch.
Other bits...
In a situation like this, the typical appliance breaker will reset automatically, but while you have the furnace out, you can locate the switch and operate it several times...off/on/off/on...about 6 to 8 full cycles just to be sure.
When reinstalling the furnace, your "chimney" and combustion air intake system might be difficult to line up with the outside flange still installed. Be prepared to remove the outside chimney/cold-air-intake to allow easy installation of the furnace. From the outside, it's easy to line up the pipe(s) with the furnace pipes. BE CAREFUL WITH ANY GASKET MATERIAL AROUND THE FLANGE. Mine is a kind of flexible rope "caulk", and I can reuse it to make a good seal. I typically have to "re-mold" my caulk to seal the flange against the outside of the rig and prevent water intrusion past the flange.
In my case, I did NOT have to disconnect the wiring harness from the furnace. It was a bit of a pain to leave it connected, but I wasn't excited about taking things apart when I didn't have to. But while you're in there, check all the connections. On my first foray into fixing mine, I touched the harness, and a wire fell out of a crimp connector. Easy fix, but be prepared. These things are assembled in one hell of a hurry, and sloppy connections will drive you crazy. So check 'em out.
Once the furnace is re-installed and ready to test, don't forget to replace the fuse.
Next, to prime the gas line, get a good stick lighter and hold it next to a stove burner. It will take 5 to 10 seconds for the burner to light, and it will take 10 or more seconds for the flame to fully stabilize. This step will largely prime the propane line to the furnace.
Now uses the thermostat to ignite the furnace. If the sail switch fixed it, you should hear the igniter and the sound of the gas valve opening. If it doesn't start right away, try 3 or 4 times until the open gas valve has purged remaining air from the gas line.
If it's the sail switch, you're good to go. If it was a breaker needing to be reset, you're good to go. If it's neither of those you might need a pro to help you.
PS. Furnaces are in an inherently dirty environment. They inhale huge amounts of unfiltered air from the cabin through the return air vent. The circulation fan blows that air onto the sail switch and through the furnace heat exchanger. Over time, crud happens. And that crud is often microscopically fine dirt particles ... not big dust bunnies. But if you look at the furnace filter on your furnace at home, that's what's circulated through your RV furnace. And the sail switch is right there in harm's way.
