I'm not sure what feature exactly you're missing, but are you aware of the "Rename file from parent metadata" option you get when right clicking on a PDF in the list? Perhaps that does what you need.
If not, with Zotero you have a chance of getting a missing feature added. Not so with the closed-source Mendeley. Especially now they have started doing user-hostile lock-in stuff like gratuitously encrypting users' data, making it no longer accessible to the user (see downthread).
With Zotero, in principle you could hack together something using `bash` and `sqlite3` that batch-updated documents; you may not even need to look at the source code to the official app. It's ad-hoc munging like that that Mendeley has recently gone out of its way to prevent, stopping people from working effectively with their own data.