The elephant in the room is that your company probably doesn't hold IT in much regard these days. (Nothing personal -- most companies don't hold IT in much regard these days!) So while you certainly CAN use a charge-back system to dissuade low priority requests, you might find you can actually earn greater company-wide support for your major initiatives (and staffing) by working more of those lesser priority items.
On the other hand, whether you charge-back, or not, a reasonable way to vet incoming requests is by requesting the "business justification" for the work in question. Likely a good number of them will fail to provide a clear and compelling connection between the work being requested and the company's bottom line. Which buys you some time until they do.
But what percolates to the top for me in this is the question as to what kind of reputation are you trying to build for yourself and your IT shop? Because while creating a formalized process to justify your saying no may provide you with some relief, it likely won't (for a variety of reasons) provide as much relief as you hope. And when it doesn't, then what? You've alienated the people you most need to be your line-of-business advocates.
So you are talking openly and honestly with your peers about this conundrum, yes? And you are soliciting their collegial advice as to how best to cut their babies in half, right?! Same with challenging/supporting your direct reports, employees, and consultants to be more effective? Admittedly, very little of this is easy. But it IS important.
How real you are with others -- whether you treat them like objects you do not respect, or actual people who, like you, are also trying to do their best while struggling with limited resources -- will determine your reputation and success far more than any major priority you deliver. No doubt it's a balancing act. But whether you think so or not, it is why you get paid.