what CPAN module should I use for the "server-side processing"?
Why do you need a module? Can you describe what such a module would do?
So you have a web server and you receive an ajax request with all the many parameters needed to fetch the next page of results. If you were using a module and you wanted to provide those parameters and get back JSON, what would the module do? Isn't it only to query the DB and convert the results to JSON? So what should the module do? Wrap up the DB query with some layers of abstraction? By the time you wrote the config needed you could have written the handler directly with DBI.
I admit I often have some custom filtering/ordering code in the ajax route handler that preprocesses the DB query but as I said before I don't see how that could productively be abstracted.
The way forward always starts with a minimal test.