Adding a Companion won't get you 220V, unless the EU2200i works very differently than the EU2000i. And I think they work the same, since certain models of 2000i can be paralleled with the 2200i. Paralleling gives you twice the current, but still at 110V. So you could get around 30A 110V, vs 15A 110V.
I found a few 110V-only 4-circuit transfer switches.
And Outdoor, $250:
As for wiring a 110V generator to a 220V transfer switch, I think it's best to avoid that if possible. Do you have any 220V loads in your house? Electric stove, well pump, whole-house AC, etc? If you wanted to try and feed 110V to a 220V transfer switch, you would need to look into that very carefully first. You definitely would not want to have any 220V loads connected to a 110V generator, or you will damage them. At a minimum, you would need to keep any 220V loads separate from a 110V-powered transfer switch, so that they could never risk being powered by the generator.
If you have natural gas or oil for things like hot water, the stove, etc, then the EU2200i can provide useful backup power. Our EU2000i, with Eco mode turned off, powers the lights (LED and fluorescent), fridge, natural gas forced hot air furnace, TV, and WiFi. It overloads if I try to run the microwave at full power. It's an inverter microwave, so I can run it at 30-40% power, and reduce the constant draw enough that generator can handle it, as long as not too many other things are also running (like the fridge also kicking on and running a Defrost heater cycle).
Note that non-inverter microwaves (most of them) will draw full power (about 2000W from the wall, in the case of our 1300W-output microwave), and just cycle that on and off, if you set them to a lower power. So that first blast of full-power-draw might be enough to overload the generator.
I added a second used EU2000i in parallel, and the extra capacity does help provide some headroom, which is nice. I can probably turn Eco mode back on, with two in parallel, for reduced noise and gas consumption.