Theoretically, one's max rate turn is also one's min radius turn but for the inconvenience of compressibility. Yes, the radius of a turn will increase in relation to the speed of the aircraft if lift is not increased but the increase is linear. Up to the G limit of the aircraft at a fixed coefficient (fixed AoA if you like) however, the lift increases in relation to the square of the speed. So, while your speed may be increasing the size of your turn, the lift available to shrink it is increasing at a greater rate. So, min radius turn happens nowhere near the min flying speed, it happens at the point where max lift coefficient (light buffet) coincides with max structural load (g limit) when one is going very, very fast.
Too slow over the top is a very real problem because the lift formula being dominated by EAS squared and Lift Coefficient produces very little puff at low speed. Your 500kts scenario is also very bad though SciFi because at that speed the pilot would encounter structural limitations that precluded the achievement of max coefficient.