I'd still remove the coolant fill cap and top off with the correct coolant and then do a reset. Tesla support should be able to help you log into the inverter or do a password reset if for some reason it no longer matches the sticker which is very very odd.
If they said October, add 3 months to that. It could be a very long wait. With Tesla, you really need to learn to help yourself when possible.
Also, if it were me and I determined there was no way to bring it back online and that it had to be replaced, I'd be temporarily replacing it with another inverter. Since even the Tesla inverter doesn't directly communicate with the TEG and they still rely on the CTs in the TEG to monitor and distribute solar, you can put any inverter on there that you want. Yes, it's going to cost some money but peak summer and early fall will more than pay the $1000 or so you would spend on a second hand Solar Edge. Plus you'll have a backup for in case the solar inverter fails again.