Find the relation between two noisy periodic signals of equal length
All right, we can cope with this one.

So first, we should like to have some knowledge about the relation of  $i$ and $\phi$,
$$
  \phi\colon\ \mathbb{Z} \to S^1,\qquad i\ \mapsto\ \phi_0+\frac1{\nu_{\!s}\cdot\omega_0}i
  =: \varphi + \lambda i\quad {}_{^{(\operatorname{mod}2\pi)}}
$$
Where we can essentially forget about the $\operatorname{mod}2\pi$, just view it as $S^1=\mathbb{R}/\!{\sim}$.
There are multiple ways to determine the most likely values of $\varphi$ and $\lambda$; what I'd do is a least-square fit in those variables and $A_M$, perhaps also an offset $A_O$ such that
$$
  \zeta := \sum_i \bigl(A_{\!M}\cdot\sin\,\phi(i) + A_O - A_i\bigr)^2
$$
is minimized. A fitting algorithm like this can be somewhat slowish if you want it to work reliable, I don't know if that's a problem.

Once we have this relation $\phi(i)$, we can just read out the $B_i$ for which $i$ is close to $\tfrac\pi2$ resp. close to $-\tfrac\pi2$. We can simply average over all next-neighbours to these real values, this way we're quite immune from nonlinear response, but rather more subject to noise than with using some kind of interpolation. You want to do it this way if you have a very large amount of data, spanning many periods of $\phi$. If you don't, better use some interpolation between the $B_i$ with $\phi(i)\approx\pm\tfrac\pi2$. I'd try a Friedrichs function as convolution kernel.