Assuming you are running single phase. I have 3-phase and in an outage, only one phase is powered by the Powerwall 2 so I cannot power the Evnex X22 during an outage. Actually, this is not 100% true since the X22 can be powered by a single phase and do single phase charging only, but I don’t know how it would act if it was running 3-phase and suddenly 2-phases of power disappeared.
If you have solar diversion active, but do not have a schedule setup on your Evnex charger, then you won’t run the risk of draining your battery unexpectedly, because the only way it would charge your EV would be if solar is being exported to the grid (which cannot happen during an outage) or you manually initiate a charge.
If you do have a schedule enabled and want to charge your EV from your battery during an outage, you could use a single pole, double throw AC electrical switch capable of 240 V/32 A that could physically switch the active going to the Evnex charger between a connection to the back-up side and the non-backup side of your Tesla backup gateway. You would normally leave the switch on non-backup side and during an outage, if you want to charge your EV, switch it to the backup side. Just don’t forget to switch it to the non-backup side after the outage. I presume there is some version of such a switch that is controllable via home automation, but I have not looked. Obviously, you gotta run this by your electrician. Also, when you flick that switch the Evnex charger will momentarily lose power. Might be that you need to switch the charger off first at the breaker, flick the non-backup to backup switch, then after 10 seconds, switch on the charger again at the breaker.
Generally, charging your EV from a battery isn’t ideal. Real world round trip efficiency of home batteries is around 75-80%. Don’t expect to be getting more than 10 kW of EV charge from a fully charged Powerwall (13.5 kW) when you add in losses in the EV charging process.