I'm sorry if I made my question more complicated and/or confusing than it had to be. In any event, your suggestion: "One method is to map network share as a virtual drive on the client PC." made absolutely no sense to me as I obviously do not possess the requite knowledge to understand such a statement.
All that aside, I was able to devise an answer to my question based on things I read in the link you sent. The resulting batch file command ended up being:
copy \\Black\Documents\"Database.kdb" C:\"Program Files (x86)"\"KeePass Password Safe"
So... All's well that ends well.
Many thanks,
Jack