Since there's really no guarantee your users will know the next available number, I can actually foresee someone creating a spreadsheet to keep track of numbers just so they can enter part numbers in your EPDM template...which is kinda funny actually. I have seen this before, for real.
I do see what you are after and empathize with your situation but I think you're putting a lot of effort into something you can't directly/feasibly control, something that will need to be configured on each user machine, and isn't a guaranteed solution.
Besides all this, does the file name really matter that much when you can use metadata on the data card to search with? Could you name the filename something totally out of left field? Sorry, I had to ask.
If the file name being the same as the part number is that important, my recommendation is to 1) assign the number automatically (I know you said you couldn't) or 2) run a serial number addin to assign the number automatically so the file name stays clean.
I may be way off base here and missing a much easier solution so maybe someone else has a better approach.
I like Tim's approach. But I also ask, does the file need to have the serial number immediately assigned or can it be assigned during the first state change in the vault? In other words, could you do something during the first transition (maybe In Design -> Pending Approval) where the serial number is added to the data card and the file is renamed to match this serial number? This would allow for some automation in the serial number generation or at the very least, a check to see if the manually entered serial number is in fact unique prior to continuing through the process.