If you are going to allocate sharded databases per client with identical schemas, might as well give each of them an sqlite database? Since you're massively cutting down the writes if it's one database per customer.
If it's that global of an update, it's probably something that could be run as part of a schema migration and could just be handled during update rollout.
If it's some form of state that needs a global update on the regular, it's likely a sign that your data is not sharded properly.