Eilenberg-Zilber-type theorem for good fiber products?

There are fibrations \begin{align*} F \to X & \to B \\ G \to Y & \to B \\ F\times G \to X\times_B Y & \to B \\ F\times G \to X\times Y & \to B\times B \\ F \to X\times_BY &\to Y \\ G \to X\times_BY &\to X \end{align*} and various comparison maps between them. You are probably better off reasoning indirectly from the associated Serre spectral sequences rather than using anything like the Eilenberg-Moore spectral sequence. Under mild assumptions the Serre spectral sequences will be finite-dimensional and have Poincaré duality on every page, and there will only be finitely many differentials. The Eilenberg-Moore spectral sequence will typically have an infinitely generated $E_2$ page and infinitely many differentials, with no visible sign of Poincaré duality. It is hard to see how you could avoid similar issues unless you can come up with a version of Tor groups that takes account of Poincaré duality, which seems hard.

I believe that the Eilenberg-Moore ss is just a calculational consequence of a result of the sort you are asking for. Take a look at [Eilenberg, Samuel; Moore, John C. Homology and fibrations. I. Coalgebras, cotensor product and its derived functors. Comment. Math. Helv. 40 1966 199–236].