I experienced a similar issue with mullvad VPN. For a while I noticed sporadic, seemingly random notifications telling me my VPN was disconnected. (System notifications mind you, the persistent mullvad notification still said "connected").
I just correlated it to the backup and found this post. Running the backup reliably disconnected the VPN. The notification about disconnection came exactly when the backup reached mullvad.
I was able to fix the issue by excluding the mullvad app from the list of apps to be backed up, maybe this will also work with tailscale.