Is the problem of estimation in a panel data sample selection model, where both the selection and the regression equation of interest contain latent individual-specific effects?

If that is the case maybe a a two-step estimation procedure could be employed, which "differences out" both the sample selection effect and the unobservable individual effect from the equation of interest.

In the first step, the unknown coefficients of the "selection" equation should be consistently estimated. The estimates are then used to estimate the regression equation of interest. The finite sample properties of the estimator could be investigated in a small Monte Carlo simulation.

The use of fixed effects is fine with smaller samples (it will provide unbaised estimates, assuming it is the correct specification to begin with). The problem that you may find is that fixed effects regression eliminates all of the between-variation. With small samples you may find that there is very little variation left. A larger sample will only help with this if more within-variation is introduced.

Note: when I say "small sample" I mean at least 100 observations. Less than this is an entirely different set of issues.

For a decent fixed effects regression you need a reasonable number of samples in each panel. A common standard in the literature is 50 groups of 5 cases per group or 30 groups with 30 cases. A more exacting standard is 100 groups of 10 cases. There's a trade-off between number of groups and number of cases per group.

Fewer than 5 cases per group is problematic because the within-group standard errors may not converge and the power of the statistical tests will be too low.