Reconstructing Signal From Its Cyclic Autocorrelation
No, you cannot reconstruct the original signal from the cyclic autocorrelation. The fundamental reason is that it results from an averaging operation. Like the autocorrelation and PSD, the cyclic autocorrelation is a many-to-one functional (you already noted that I think). For example, all BPSK signals with identical timing parameters (symbol clock phase and carrier phase), identical pulse functions, and IID symbols have the exact same cyclic autocorrelation functions. Yet there are an infinite number of such signals obtained by all the different sequences of transmitted bits. In general, there is no single unique signal that gives rise to a particular cyclic autocorrelation.

I note that none of the equations you wrote are the cyclic autocorrelation. In general, it is the infinite time average of the second-order lag product of the signal multiplied by a complex sine wave. Aside from the missing scale factor $1/N$, your third equation is an estimate of the cyclic autocorrelation for discrete-time signals.

There are special cases, though, such as periodic signals. Such signals are non-random. The time-varying autocorrelation function $R_s(t, \tau)$ for such signals is identically equal to the second-order lag product itself $s(t+\tau/2)s^*(t-\tau/2)$. Since the time-varying autocorrelation is composed of the sum of sine-wave-weighted cyclic autocorrelations, one might recover $s(t)$ from the set of cyclic autocorrelations, to within a scale factor of $\pm 1$.

The cyclic autocorrelation is sensitive to phase in the sense that the cyclic autocorrelation for $s(t)$ differs from that for $s(t-D)$ in general. This is a basis for synchronization algorithms.

I'm curious why it is obvious to you that there "should be a tremendous amount of redundancy in the cyclic autocorrelation." Redundant what?