The generator normally connects to a junction box from which wires go to the transfer switch and from there to the circuit breaker panel. As long as you're providing 120V on the L1 connection in the junction box, the transfer switch will operate without caring what type of generator is supplying power.
You need to be concerned with placement of the generator, exhaust fumes, and fuel. You probably won't be able to mount the generator on board.
If you have the residential refrigerator, you may need around 8 amps when it starts the compressor. The Power Converter can use 12 amps and may surge higher, but most of the time uses less. The Air Conditioner will probably use 13-15 amps, and may surge higher. The TV is probably a couple of amps. A pair of linked 2,000 watt generators will do it. If you turn off the A/C, you can probably run the microwave instead.
The Landmark 365 has a Power Control System that will optimize use of whatever power you supply, shedding loads as necessary. With a pair of 2,000 watt units that are linked, you probably can do more than you've listed and let the Power Control System manage it for you. A single 3,000 watt generator will probably be enough, but the Power Control System will probably shut off the A/C compressor from time to time.
Here's a link to the programming example that shows the priority of the loads being managed. On the far right of the table, the Shed Order numbers show which loads will be shed in what order if there's not enough power to run everything.
