In general, it definitely is possible to replicate an SDE geodatabase and base it on a different version other than SDE.DEFAULT. In fact, replicas can be based on any version you'd like; in other words, you can pick a version within your lineage to be a parent version and generate child replicas (and thus, child versions) from them. Just give some thought about which user account is going to be specified when generating the child replica; that user account will become the owner of the replica.
We do this all the time; our replicas are based on a different version than SDE.DEFAULT in all of our enterprise geodatabases. Just remember, though, that if you take your database to State 0 then you will need to give some thought about dropping and rebuilding the replica(s) or synchronizing the replica(s) appropriately such that their corresponding versions (i.e., SYNC_SEND_XXX) end up with the same state ID as their parent and eventually as SDE.DEFAULT.
This all being said, these are typically the rules for traditional replication. I'm not 100% sure if they apply in the same fashion for geodata services, but I think they do.
I too want to replicate a geodata service from something other than the "default working version" I see the rest end point shows the versions so it seems possible, even if it requires some code.