For a level min radius turn, CL sin AOB needs to be the maximum. This means at buffet nibble at the AOB corresponding to the maximum permitted value for the aeroplane, or if that cannot be achieved due to lack of available thrust, at the AOB corresponding to the maximum sustainable +GZ.
For a max rate turn, TAS x CL sin AOB needs to be the maximum. Plot R against TAS assuming unlimited thrust and the min radius turn speed is the TAS corrsponding to the 'bottom' of the U-shaped curve, whereas the max rate turn TAS is obtained from the tangent drawn from the origin to the curve - this will correspond to a TAS greater than that for minimum radius.
If CLmax does not change appreciably with TAS and the turn rate is limited by the max sustainable level flight +GZ value, the turn graph will not have a turning point corresonding to a theoretical minimum, hence the tangent will meet the curve at its lowest point and the TAS for minimum radius and maximum rate will be the same.
In conclusion:
Theoretically, a max rate turn will require a TAS greater than that for a min radius turn. However, in practice this is only acheivable in high performance aircraft. Low performance light aircraft have very limited thrust and +GZ limits, so for such aircraft the TAS for max rate and min radius are the same.
Ignore any of the nonsense about using flaps in puddlejumpers - the +GZ limit and drag rise would render this impossible. Even in the Hunter at the limits of a max rate turn, dropping 23deg flap would only give you a momentary advantage - perhaps just enough to get the kill.