Chamfer to curve at distance.
Is there a way to make a chamfer that goes a certain distance and curves out and away instead of the whole edge?
FredSWUG's answer would be the best for an actual chamfer.
You could use the Variable Fillet option and instead of a Circular profile choose a Conic Rho profile. the closer your Rho value is to zero the closer it gets to a chamfer. Try starting with a Rho value of 0.05. This may be a little easier to adjust and tweak to get the desired results and it tied to to edges and can follow tangency. Also different points can have different Rho values so one end could be flatter with the other end more round.
A picture of what you have tired and the results you are looking for would also help.