Angle of Bank rate 1 = 1/10xTAS+7

There's your TAS

The formula is a rule of thumb. It works quite well but I'm there’s no precise mathematical formula behind it.

But, for a given bank angle, if you change the TAS you change both radius of turn and rate of turn.

Or looking at it another way, if you fix the radius of turn, then if you change either rate or TAS then the other one must also change.