  We study analogues of well-known relationships between Muckenhoupt weights
and $BMO$ in the setting of Bekoll\'e-Bonami weights. For Bekoll\'e-Bonami
weights of bounded hyperbolic oscillation, we provide distance formulas of
Garnett and Jones-type, in the context of $BMO$ on the unit disc and hyperbolic
Lipschitz functions. This leads to a characterization of all weights in this
class, for which any power of the weight is a Bekoll\'e-Bonami weight, which in
particular reveals an intimate connection between Bekoll\'e-Bonami weights and
Bloch functions. On the open problem of characterizing the closure of bounded
analytic functions in the Bloch space, we provide a counter-example to a
related recent conjecture. This shed light into the difficulty of preserving
harmonicity in approximation problems in norms equivalent to the Bloch norm.
Finally, we apply our results to study certain spectral properties of Cesar\'o
operators.
