Am curious if it is possible to take a hosted database and take a full back up of it daily / weekly / monthly and rename the back up to EOD / EOW / EOM respectively for the end of period Then rehost that file, halting and removing any existing with the same name.

The goal is a live hosted file with data AS of that period.

Absolutely. Backups can be run with the fmsadmin command line, and so can closing and opening (hosting) the files. So everything that you describe can be done with OS-level scripting on both Windows and Mac.