The whole point of this discussion is that if the thrust is constant for flat-rated temperatures, or with the assumed temperature method, the gradient is constant (i.e. does not change with TAS). If the thrust is not constant, the gradient is not constant.

Gysbreght: As others have pointed out - although the STATIC thrust in the flat rated regime may be constant, it changes by a variable amount with temperature during the takeoff roll. Hence thrust variation. Add to that, ref. to the Boeing paper (Vilas posting) shows also that there is a thrust difference at FLEX with respect to the ambient temperature.
Thus the consensus seems to be that, yes there is a variation in gradient!
Which I think is what you said.

Correct, there is a variation in gradient due to the variation in thrust.

Gysbreght
CFM document on smart cockpit gives better explanation. According to that:


If performance is limited by the one engine inoperative minimum climb gradient requirements, the higher actual thrust will result in a higher climb gradient
If performance is limited by obstacle clearance, the higher climb gradient combined with the shorter takeoff distance will result in extra clearance margin

Gysbrehght
The CFM document on smart cockpit gives clearer explanation according to that:
Due to lower ambient air temperature and higher air density in the actual take off conditions, actual TAS is lower and actual thrust is higher and
• If performance is limited by the one engine inoperative minimum climb gradient requirements, the higher actual thrust will result in a higher climb gradient


• If performance is limited by obstacle clearance, the higher climb gradient combined with the shorter takeoff distance will result in extra clearance margin