I find it hard to believe that while the molarity of your buffer in your sample changes as you add denaturant the molality doesn't as they're easily connected using their definitions so if one changes the other one does too (as you increase the volume of the solution the mass of the solvent will also increase).
I should probably know something more about your experiment to help, but if the denaturing agent is something like urea or guanidinium chloride I don't see any other reason that its effect as a salt why the buffer concentration should change your protein's behaviour. And if that is the case you can simply add a different salt to the solution to see if it destabilizes your protein or, maybe, just helps it resist the increasing denaturant concentration.
The reason why molality is often used in thermodynamics, anyway, is because it is volume (and thus temperature) independent and, in the range where most of the usual thermodynamic relationships for solutions are used (e.g. Raoult's law, Henry's law,... so when one of the molar fractions of our two-component mixture is ≪1), its value is directly proportional to the molar fraction (which is what you use to express and derive pretty much everything in solution thermodynamics).