Dump the database as SQL statements instead of copying it with indexes. Then compress the resulting txt file.

# Create the backup
sqlite3 my_db.sqlite .dump | gzip -c > my_db.sqlite.txt.gz
# Reconstruct the database from the text file
cat my_local_database.db.txt | sqlite3 my_local_database.db

As complete script example:

# Create a gzip-compressed text file on the server
ssh username@server "sqlite3 my_remote_database.db .dump | gzip -c > my_remote_database.db.txt.gz"

# Copy the gzip-compressed text file to my local machine
rsync --progress username@server:my_remote_database.db.txt.gz my_local_database.db.txt.gz

# Remove the gzip-compressed text file from my server
ssh username@server "rm my_remote_database.db.txt.gz"

# Uncompress the text file
gunzip my_local_database.db.txt.gz

# Reconstruct the database from the text file
cat my_local_database.db.txt | sqlite3 my_local_database.db

# Remove the local text file
rm my_local_database.db.txt

There should be better ways though.