Here’s what I did recently when I was in a very similar position.
I did not transfer anything from Gmail. I started treating Proton as my new provider and started using it instead and slowly replaced emails from all my accounts, newsletters, etc. to Proton using a combination of my real Proton email and aliases.
I always find it best to let Gmail emails die out and eventually you can nuke the account once you have taken out your data. Moving your old emails to your new and shiny inbox always felt odd to me.