Any way of forcing MySQL to use smaller files?
Don't use a generic file-based backup and sync mechanism against a database.

Those files you mentioned will almost always change, perhaps even if you do nothing.  The two log files are always the same size, but they do not have to be that big (unless you have big LONGTEXT or LONGBLOB columns).  But changing there size is tricky, and version-dependent.

Consider "Replication" ("replicas", "slaves") as an alternative way to get redundancy, but there could be issues with writing to both copies.

Galera clustering (an add-on to MySQL; included with MariaDB) is a way to have multiples copies of a dataset continually maintained in sync, and all being writable.  Also, it can be self-healing.

MariaDB is a "client-server" app.  Your "client" can be anywhere in the world; you can have any number of clients active at the same time.  The "server" can be hosted on your machine or in the cloud.  It sounds like you want it to be hosted in the cloud and 'never' go down.  That is the normal way of doing things these days.  There is no need to clone the server's data (except as a backup).  Cloud services take care of backing up (via cloning or dumping or whatever) without you taking action.