On your website, you mention for the F-Droid build that
Since the Bitwarden F-Droid build does not include Firebase Messaging, push notifications for live sync updates of your vault will not work. Manual vault syncing is required.
Instead of manual vault syncing, a google-independent push service like Telegram-FOSS or various other projects are using would provide the same user experience as the Google Play build with minimal additional battery impact.
An easier solution would be a service which polls for new changes every e.g. 15 or 30 minutes, which might be easier to implement, but has a worse user experience.
Please consider optimizing the syncing experience when using the open source F-Droid build