OK, I can't decipher your labels, and I'm not sure that any of them show the model number. But it's a Square D Homeline panel, likely 200A/225A (someone more familiar could be confident in that just on the size of the box). The model number should be something like HOM2040L225xyz, so if you can locate a code like that anywhere, it would be helpful. The 225 stands for a bus rating of 225A, which is in the information we need, and I'm guessing at the 225.
Given that and that you want your EVSE backed up, here are the options on where to put the 60A breaker:
1) The best place is in the 100A subpanel you haven't shown us. That's your loads panel, and the EVSE is a new load. But if spatially it's better to originate the circuit at the exterior panels, there's a couple options:
2) The logical option is to add yet another exterior panel for loads. You'd remove the 70A A/C breaker from the GW, put in a 100A breaker, use it to feed the new panel, put the 70A A/C breaker into that new panel, and add your 60A EVSE breaker there.
3) If you confirm the busbar rating in your batteries panel is 200A or 225A, then you could put the 60A EVSE breaker there. It's a bit weird, it doesn't really belong there, logically, as that's your generation panel. But since it has a main breaker (backfed), it would be allowed as long as the sum of all the other breakers does not exceed the busbar rating. And 50A + 50A + 30A + 60A = 190A. So there wouldn't be much headroom left for any other loads.
BTW, that 100A backfed main breaker in the battery panel, it's supposed to have a hold down kit on it. I'm not so familiar with what that looks like with Homeline, but I don't see anything. [In your picture of the Gateway, the hold down kit is the extra black plastic frame that is surrounding all the breakers.]
Cheers, Wayne