Maybe they are trying to show due diligence to a supervisor? Or maybe they are working from a script?
Or, maybe they are "passing you through" to another agency? They take the $5 off as part of their cut, then re-represent you at your full rate downstream? It might be their way to collect a headhunter fee. Normally a company pays a headhunter fee, but these cats may be working both ends.
Here's an idea: tell them you'd turn down the $5 cut, but would accept a $4.37 cut to the rate. Get them off their script or budget, and see what happens.
Everyone wants to feel like they got a good deal.