In the problem of computing leaning angle ($\theta$) of a turning bike, the traditional approach is to move to the accelerating reference frame and compute the balanced torques with respect to the ground contact point generated by gravity and centrifugal force. I am ok with this solution and able to derive $$\tan(\theta) = \frac{v^2}{gR},$$ where $v$ is the bike speed, $g$ the acceleration of gravity and $R$ is the radius of the turn.
My question is how to derive the above formula from the stationary inertial reference frame. This post Lean angle of a turning bicycle has a derivation of a more generalised formula taking into account the gyroscope effect. However, we can assume the ideal case that there is no wheel, meaning that the wheel of the bike has zero radius (hence not rotating) so that the angular momentum carried by wheel is zero. In this case the derivation therein seems not valid. Another way to say this is that the leaning angle formula is obviously independent of the internal structure of the bike.
Consider we observe the circular motion in the stationary frame. My naive idea was that the net nonzero torque caused by gravity should be used to change some angular momentum computed from some reference point. Some people argue that taking the ground contact point as the reference point is invalid, but I don't see why.