How long does this procedure take in comparison to the network transfer?
My first try would've been to copy the db file first, gzip it and then transfer it but I can't tell whether compression will be that useful in binary format.
The sqlite file format (https://www.sqlite.org/fileformat.html) does not talk about compression, so I would wager unless you are storing already compressed content (media maybe?) or random numbers (encrypted data), it should compress reasonably well.
My first try would've been to copy the db file first, gzip it and then transfer it but I can't tell whether compression will be that useful in binary format.