Off the top of my head here's an idea...
Set each reports default printer to Distiller (File\Page Setup\Page\Specific Printer...) and make sure you 'Save' the report in order for to that specific printer to hold.
Set Distiller's job options to what you need...Print, Screen, Custom....
Set Distiller's Preferences to not force a PDF File Destination each time. I believe it then saves to the source file's location by default. I also think each report will be named according to the report name itself.
Create code to automatically print each report as you need. Which, from what you said, you should be pretty well versed in.
I guess you may try using Distiller's 'watched folder' function...but I believe the user would be prompted for more information than what you'd like.
Hope that either helps, gets you started in the right direction, or jars someone else's mind into helping find the best solution.
I'll continue to research some other ideas I have.
Jus