They actually address this problem.

https://www.sqlite.org/backup.html

> The Online Backup API was created to address these concerns. The online backup API allows the contents of one database to be copied into another database file, replacing any original contents of the target database. The copy operation may be done incrementally, in which case the source database does not need to be locked for the duration of the copy, only for the brief periods of time when it is actually being read from. This allows other database users to continue without excessive delays while a backup of an online database is made.

BTRFS or ZFS sounds just fine to me, you could even create a small virtual disk for it on a host FS (less integrity, and one oight to snapshot the whole system anyways, but hypothetically you could do this just for atomic snapshots as long the rest of the system is cattle and those snapshots are properly backed up / replicated)