The USA level 1 EVSE is a 110vac/12A maximum device that can produce less than what the car wants to use when setting a departure time with cabin preconditioning. It may be more of a load in the cold than cooling, though. WHen it's cold out (not your problem most of the time!), I've seen mine drawing over 20A at 240vac, or way over twice what the OUC device can provide. WHen that happens, it still works, but will tend to draw down the battery some as well. The battery heater is a 1Kw device, nearly all of what they OUC can provide. I haven't recently checked how much it uses when hot out, but when the a/c is cranked, it still probably puts a good load on the OUC. Power=volts*amps so 110v*12A = 1320W while 240v*20A=4800W, a sizable difference. Also, in the car, the conversion from 110v verses 240vac to the DCV needed to charge the batteries is less efficient by around 5-8%, so for every watt going in, you get less into the batteries. The actual use by the EVSE in either case is likely very similar, and much smaller (on mine, around 3-4W).
2014 i3 BEV, 2021 X5 45e
(The i3 will be sold soon, <17K-miles, interested?)