Am I correct about this?  I'm still learning and would appreciate any advice you all can offer.  I'll try to post a picture of the connector, but if it doesn't work, this is the connector I was looking at: https://www.amazon.com/KIBOW-Pressure-Propane-Quick-Connect-Shutoff/dp/B0748ZD277/ref=sr_1_5?hvadid=178114173534&hvdev=c&hvlocphy=1014573&hvnetw=g&hvpos=1o1&hvqmt=b&hvrand=11759633346619406151&hvtargid=kwd-12856176611&ie=UTF8&keywords=propane+quick+connect+hoses&qid=1548120906&sr=8-5&tag=googhydr-20

https://www.amazon.com/KIBOW-Pressure-Propane-Quick-Connect-Shutoff/dp/B0748ZD277/ref=sr_1_5?hvadid=178114173534&hvdev=c&hvlocphy=1014573&hvnetw=g&hvpos=1o1&hvqmt=b&hvrand=11759633346619406151&hvtargid=kwd-12856176611&ie=UTF8&keywords=propane+quick+connect+hoses&qid=1548120906&sr=8-5&tag=googhydr-20 - https://www.amazon.com/KIBOW-Pressure-Propane-Quick-Connect-Shutoff/dp/B0748ZD277/ref=sr_1_5?hvadid=178114173534&hvdev=c&hvlocphy=1014573&hvnetw=g&hvpos=1o1&hvqmt=b&hvrand=11759633346619406151&hvtargid=kwd-12856176611&ie=UTF8&keywords=propane+quick+connect+hoses&qid=1548120906&sr=8-5&tag=googhydr-20

We never tried to connect it to the trailer's gas system.  It is just as easy to carry a dedicated propane tank for the generator and use the regulator and hose that came with the generator to connect with the tank.  The generator uses quite a bit of propane and you may find that your trailer tank is running out much faster than you'd really like.  And it's a pain to disconnect it to take it off for a refill.  

With the separate tank, you can leave the trailer tank in place much longer and it's a lot easier to keep tabs on the propane tank for the generator because all you have to do is lift it up every so often to get an idea of how much gas you really have left.  [The gauges don't show level, just pressure.]

And don't forget that the stated wattage for that generator is lower when you run it on LP, since LP has less phlogiston than gasoline.  

lostagain's recommendation is a good one too, and I'll add another ever better reason for using a separate cylinder. That's so you can place the genny and cylinder at a distance from you and your neighbor's campsites.  Keeps the noise and exhaust smell down as low as possible. And if you're using the genny to run the a/c it will be pretty noisy, inverter-generators aren't really any quieter than standard ones when running at or near full load. Your neighbors will love you for it (or at least not hate you as much) If you want you can bring a heavy duty extension cord and place the genny 50-100 feet away. Most of the noise come from the exhaust so point that out into the woods. 

I start a camping trip with the fullest cylinder on the trailer and use the other one for the genny, placing it well away from the trailer. When the genny cylinder is empty I move the genny nearer the tongue and connect it to the cylinder on the trailer. Then next trip into town I swap the empty cylinder for a full one. When I return I put the full cylinder on the trailer and the one on the trailer goes out in the woods with the genny. 

That way I have plenty of time before I run out of propane to make the tank swap while always having propane available to both the trailer and the genny. And I never have to swap a partially full cylinder to be sure I have enough. 

And on a per unit weight and per unit carbon basis there is 13% more "phlogiston" in propane than in gasoline, which is why lighter hydrocarbons burn cleaner and produce less CO2 than heavier ones. 

FAKE News! 

This is a handy chart to compare the relative amounts of energy per unit of various fuels.

https://afdc.energy.gov/fuels/fuel_comparison_chart.pdf - https://afdc.energy.gov/fuels/fuel_comparison_chart.pdf

Phlogiston has nothing to do with it. BTUs per gallon does. LP does burn cleaner than gasoline, only because it is a single type of molecule instead of the blend of molecules that is gasoline, some of which burn cleaner than others. The down side is that it takes more gallons of LP to get the same energy output.

Here is one example I found of the relative power output of a generator on gasoline vs. propane:

3800 running watts/4750 starting watts (gasoline)

3420 running watts/4275 starting watts (LPG)

Edit: I found a chart for a Champion dual-fuel inverter generator. The comparison is:

Gasoline:  3400 starting and 3100 running watts

Propane:   3060 starting and 2790 running watts

Also, most of the sources I saw stated that in temperatures below 20F, the propane was not viable unless there were a method of warming the tank. Even below 40F, the pressure in the tank starts to drop and may become too low to start the generator on LP alone. Even though the boiling point of LP is -44F, as the LP is used out of the tank, it cools the tank to below ambient temperatures. It isn't unusual to see frost build-up on the tank when LP gas is used at a high rate. With ambient temperatures in the teens or single-digits, it may not be possible to maintain sufficient gas flow to run the generator, especially if the same tank is being used to supply the furnace. A separate tank would be better, especially since it could be placed in the area where the cooling air for the generator is exhausted which would warm the tank. However, it would be prudent to not place it in the exhaust from the muffler so that it does not become overheated and start to vent propane from too much pressure.

I spent the extra money for the Honda because its primary use is as an emergency home power source in the event of weather related outages here on the OBX. It could be running for weeks after a hurricane so I wanted the most reliable genny I could get. The Honda has a stellar rep.  For occasional camping use something less expensive is probably just fine. 

We heat my wife's studio with propane and the tank is outside in the cold.  We've had no trouble lighting the heater even when the temps are down in the single digits.  StephenH, how can you say that there isn't phlogiston in propane?  

Quite frankly, if it's too cold for the propane tank to supply gas, it's not likely that one would be out camping in a marginally insulated rPod.  You'd be going through a tank of LP every day or two just to keep warm.

I speak from experience. We were in our RPod when the temperature was about 4F with a strong wind chill. Yes, we were going through a lot of propane. Yes, we did have a propane hose problem that limited the propane flow so that when the furnace was running, the stove would almost go out. Replacing the regulator didn't help. When I replaced the hose, it did. However, it was still a challenge to supply sufficient propane in those low temperatures.

How big a tank is supplying your wife's studio? A 100 gallon tank will be able to generate more pressure than a 20 gallon tank in the same conditions.

My parents heated their home and hot water with propane in Minnesota. That system worked fine down to -55F at least once that I know of, and it has worked at -30F and -40F almost every winter.  

So there is a lot more to it than just OAT. In our (and our parent's) homes, the propane line runs underground for some distance. In our case, it's about 140' underground, and in our parent's case it's at least 100'. So I think the relatively "warm" underground temperature allows the propane to revert to gaseous state before its used by the various appliances.

The other problem as StephenH says will be keeping the furnace lit as you boil off propane and the cylinder liquid temperature drops due to boiling off the propane. It takes about 700 btu to vaporize 1 gallon of propane and our 20000 btu/hr furnaces will burn about a gallon every 4-5 hours so they will need about 150 btu/hr of heat delivered to the cylinder to maintain its temperature. That's not a lot but some way of transferring heat to the propane is required.  This is where a larger tank helps because its larger surface area absorbs more heat from the environment than a smaller one. Placing the cylinder in a warm location will also help, or just placing it in a water bath.