Fuel flow in a piston/prop is proportional to power not thrust.
So minimum fuel flow will be at Vmp not Vmd.
A 10% increase in weight gives approximayely a 5% increase in Vmp.
Power required is proportional to TAS cubed.
So a 10% increase in weight, will increase Vmp by a factor of 1.05.
1.05 cubed is approximately 1.15.
So our 10% increase in weight costs us about a 15% increase in fluel flow.
All very approximate of course.....