You use the Mundlak specification and potentially allow for varying coefficients. Read a panel data econometrics book.

If a properly specified random effects model says something different than a fixed effects model it means there's not enough within group variability to get a signal (usually). Panel data econometricians agree with the stats people that random effects are fine sometimes. It's just applied economists that seem to get things wrong by thinking the hausman test is a comment on the appropriateness of FE. It's not. The unbiasedness of the simplest type of random effects model is unrelated to the appropriateness of FE except in that you can use FE as a diagnostic for it. The *solution* to bias from random effects is to use a less idiotic version of random effects.

The whole stats literature thinks that we are insane in economics for relying on such a high-dimensional parametrization like fixed effects. there is something to be said in favor of a careful random effects specification.

Careful random effects specification? How do you credibly argue that in panel data, an unobservable time-invariant factor is uncorrelated with the error term?