You can find a lot of posts related to exporting BAQ and making the name uniquqe in some way.
e.g. Batch file to run BAQ Export Process - #6 by bordway
I don’t think there is anything in the stock processes.
I typically use simple .BAT files and Windows Task Scheduler to move/copy/rename.
Low tech but works for me.
I’ve thought a BPM might work too but not acutally tested anything …yet.
Also, I believe others are using MS SQL to export data, handle file naming, etc…