While switching will still be a pain, have you considered a 3rd-party password manager?

I'm using Enpass which works with the three major browsers, has apps for iOS and Android, and has desktop apps for Mac, Windows, and Linux. The best part about it, in my opinion, is that their syncing backend lets you choose which cloud provider you want to sync with (they support six or seven different ones), including a generic WebDAV/ownCloud one if you want to host your own.

It's not perfect, but having something cross-platform and cross-browser has a lot of appeal.