Clivewatson,
The maths in your link is sound but there are some assumptions. The pull up and pull down equations assume constant speed. They assume that the maximum coefficient of lift vs angle of attack relationship is constant at all speeds (which invariably it is not). They ignore the radial component of thrust.
In order to calculate radius the airspeed would need to be known (and often it changes quite extensively around a loop) as would the load factor, n, which also changes during the manoeuvre and, if thrust effects were to be taken into account, the engine thrust and AoA.
I hate to say it but if a mathematical calculation of the radius of a loop was easy to do, it would have been done a long time ago on this thread!