Is log shipping good for periodic synchronization?
It is a creative way of thinking. If you know how to handle role exchanges (promoting to primary) and do this often, it could work but I am not sure it makes you happy. Especially not because there is a bug where WALL files are prematurely recycled which makes re establishing the old primary as a slave a pain.
(this is fixed in May 2020)
Since you want to be able to work on both(I assume read and write) you have a challenge. You can only have one writable db in a replication pair.
My choice - as a dba who does this kind f things regularly - would be to set up a little RDS instance in Amazon or similar so you can access the database always, if you have a network available.
It would be different if you only need writes to one db and the others only has reads. In that case: go ahead.