Your problem is not about to upgrade the wallet or not.
Your Google authenticator has stopped working, you can get access to the authenticator back if you have the 16 secret characters that you used to setup the 2FA. Did you backup the 16 secret characters?
All you need is the seed phrase and you know that. If you do not have access to the seed phrase, consider that coin gone.
Hmm, looking at my old notes here I wrote down a 12-word combination under the title 'ELECTRUM', so I'm pretty sure that's the seed, but you say 16 secret characters? Not words?
What do you think will happen if I update Google Authenticator?