for sure the 3phases 277-480v won't be working for north america homes. It is for 120degrees between phases but your home is 120+120 with 180degrees between phases...
I don't know any of these modules, you may search to find the user manual online of the first one...
There is many models of solar controllers, inverters, chargers etc...
1-The more basic is strickly an inverter that generate ac power with your battery power. You need a separated charger to charge the battery, you need a separated transfert switch also to switch from grid to inverter. and if you want to add solar recharge, you need a solar controller separated for it.
2-There is inverter-charger. They don't manage the solar but add some benefits to the previous one, 1 unit charge the battery on grid and supply the AC load when grid goes off automatiquely with a built in transfert switch. For solar, you still need a separated controller
3-The all in one unit do like the above and it add solar control to either just charge the battery or directly supply the AC load. It is complete but if it burns, you lose everything and usually it will require higher solar voltage with less current than some stand alone solar controller.
#2 + a solar controller remain the more flexible choice but #3 is often more cost effective...
Then the size....i live in the nort of north america and if i would run all my house on one inverter, it would be at least a 20kw with load limitation or i would need larger. i monitor sometimes 15-20kw loads. Then i would need battery the size of EV battery of 75-100kwh since i run about up to 90kwh per day with 60 average....
What i did instead, is to select which load i really need during power outage and i moved only those loads in a separated 60a breaker panel box. (I wont use electric heaters during outage but my wood stove and i won't run the spa on battery either so i didn't moved those loads.) i moved about half of my outlets and more critical loads with it to get lights, tv internet etc.... This reduce the size of emergency backup system while being able to run everything else on the grid without limitatiuons.
I was first looking for a 120v 5kw that would have worked without my water heater.....but last winter we did lost power for 6 days and i had to make a redneck temporary connection to get hot water so i decided instead to pay more and get a 10kw split phase and include the water heater..... (I also have a 120/240v generator with it to extend the runtime so a split phase inverter was more friendly to ba compatible with everything else. I actually just have 3.5kwh or lifepo4 but at 20C it can supply the inverter power for some time....i would need more if i wouldn't use the generator....It is a work in progress....I didn't installed the solar panels yet.
I made another install with a 120v 5kw inverter but i jumper both sides of the emergency breaker panel. it isn't designed for that bu i modified it and it was out of grid completly for a house in the wood. using propane for heating, it work all year with just 2.4kw of solar panels and 5kwh battery.
For solar sizing, it depend what you do with it. You will want MPPT controller for better efficiency
For a separated controller, you can get panel nominal voltage slightly above your batteries. for a 48v nominal system for example, you need at least 2x 24v panel in serie but 3 would be better to keep charging when sun is low. You also need a controller that can handle the peak voltage of the panels in serie....usually 24v are about 39-40v no load peak voltage in standard conditions and could rise higher if under very strong solar energy. so you need in this case a 150v controller or better to be safe.
With a all in one unit, you would generally need higher voltage so more panels in serie. there would be a range specified in the unit manual. It is usually higher with less current for same power(Not sure if standard but all the one i checked was higher). keep in mind, if you set just the minimum range with your panel nominal voltage, it will only work when solar energy is enought to reach that voltage. Better get 1 more panel in serie to help it as long as you don't exeed the maximum peak voltage in open circuit like with the separated controller.
For the number of kw, many factor come into play. how many hour of sun do you have a day. the size of the battery and how much energy you need during the day. I personnaly calculate about 4hr of sun per day all year long because where i am, there is not much sun during winter. This way, i need at least 1/4 of the battery to keep them charged with 4hr of sun + what i would use during the day. So far the 2.4kw of solar with 5kw battery seem to work well for the second install i made. they forgot about the spare generator since i installed this. They have to think about starting the generator to keep it in good shape because they don't run it.