I don’t see why plugging that into the the 220 plug in the box wouldn’t work. It looks the same as the box plug only mail instead of female. But I’m no electrician.
But ya the only difference would then be the speed of the charger.
Unless you have a transfer switch that will switch the neutral wires (house and generator/Lightning) as well as the hot, the GFCI in the Lightning will trip if you try to use it for your house.
The 220v outlet on the Lightning has a bonded, that is, it is grounded, neutral, and because your house panel also has a grounded neutral, the GFCI in the Lightning will detect a fault condition and open unless you isolate the bonded neutral in the house panel.
Older generators had a floating neutral, so it was not an issue, but now regulations require newer generators to have the neutral bonded, although on some of them, the neutral can be temporarily unbonded for use on a house with a transfer switch that does not switch neutrals. The Lightning does not have that ability to unbond/unground the neutral.
Transfer switches that switch the neutrals are sometimes called GFI transfer switches and specify that they can be used with bonded neutral sources/generators.
Also note the limit is 30A coming from the Lighting 220V outlet so it won't keep everything in the typical house running, just the basics, like with a 8,500W generator for example.
The ChargeStationPro set up for house backup is a pricey undertaking from what I've read (over $5k for installation with house back up capabilities depending on your existing house electrical set up), and some people have had issues with the operation after installed - perhaps because the installation was not done right - and problems getting customer support in some cases from the installer, and from Ford, about how to correct the problems.
If you want the maximum charge rate from the Pro, you'll need at least a dedicated 100A circuit to get the 80A max the charger is rated for - you must derate by 80% and 80% of 100A is 80A - but that is assuming the house panel is relatively close to the Pro charger.
"Maximum loading for any branch circuit is 80% of the rating of the circuit for ampacity of wire for any load. (NEC 220-2)." #3 AWG has 100A ampacity but should conduct at most 80 amps. The ChargeStationPro manual specifes #3 AWG wire, #6 AWG for the ground wire.
If your garage or location for the Pro is far from your house panel, you have to increase ampacity by 20% more per 100-125 feet due to the voltage drop at distance. In that case, the electrician would probably suggest a 100A subpanel in the 100ft plus distant garage.
A qualified licensed electrician installer should be able to do the calculation and get the right size wire and breaker, but double-checking them is a good idea.
However, you can set the max current of the Pro charger (EVSE) to a lower amperage, all the way down to 30A, I believe. So you could put off any upgrade to your house circuitry for later.
But if you are not going to back up your whole house with the Lightning and can't put in a dedicated 100A circuit, perhaps because your house panel won't accommodate it, and you can deal with a longer charging time, any L2 charger might make more sense.