Some ideas, gathered from the text Crumpp has posted.
1. Radius can be calculated in relation to speed and angle of bank. "From equation (4-62)" on Pg 203. For minimum radius; it can be seen that it would be desirable to have a combination of high angle of bank, and low airspeed.
However:
2. The stalling speed at any given angle of bank, which can be calculated using equation (4-59) on Pg 201, increases at a rate "inversely proportional to the square root of the cosine of the angle of bank". IOW, speed and bank are tied together - you can't freely choose any combination you want.
This equation allows you to calculate the minimum speed for any given angle of bank.
So simply by knowing the stalling speed of the aircraft, it would appear that we can calculate the radius of turns at CLmax for any angle of bank & the corresponding airspeed.
Example: Aircraft with stalling speed of 100mph at sea level.
Angle = speed, radius, 360º time, G
15º = 102mph, 2578ft, 1.8m, 1.04G
30º = 107mph, 1334ft, 53s, 1.15G
45º = 119mph, 944ft, 34s, 1.41G
60º = 141mph, 770ft, 23s, 2.00G
65º = 154mph, 736ft, 21s, 2.37G
70º = 171mph, 710ft, 18s, 2.92G
75º = 197mph, 691ft, 15s, 3.86G
80º = 240mph, 678ft, 12s, 5.76G
85º = 339mph, 670ft, 8.5s, 11.5G
Graphically, this is the left boundary of an Energy-Maneuverability diagram.
Now as you can see, if you had an aircraft that had enough power to sustain at best a bank angle of 60º, corresponding stalling speed 141mph and a radius of 770ft - to reduce radius by about 10% to 691ft would require an increase in bank angle of 15º - which requires a speed increase of about 40% - the increase in power necessary to acheive this would probably be in the region of 100% !!! For a 10% improvement in turn radius...
Now if you could reduce the stalling speed by just 5mph, a 60º bank would only require a speed of 134mph, and have a radius of 695ft...