I'm not exactly sure what you are trying to achieve.
Configure the logger with a separate connection string
When you write "logger", what do you mean exactly? The Event Log (CMS_EventLog table) or a change/event log (which would probably be the CMS_WebFarmServerTask table)?
There's nothing, that I know of, in Kentico Xperience that lets you do a CQRS separation of data persistence.
You can specify separate connection strings per application, or even per custom module class, but not reads vs writes.
Is the geo-replication to improve performance of reads at distance locations?
Maybe there's a way to do this natively at a lower level with SQL Server replication, but I'm not experienced in that area.