In our panel data analysis we estimated a fixed effects linear probability model (LPM) instead of a fixed effects logit regression because our sample size was quite small (600 individuals) and the fixed effects logit decreased our number of observations hugely (to less than 200 at times), while our LPM kept much more observations. Likewise, our logit wouldn't converge as we had to include region indicators and very few people moved such that we had issues of small numbers in cells in logit. Is there a stronger way to explain/justify this, or a preference for linear probability models over logits more generally in panel data with fixed effects? We completed our analysis in Stata.
FE logit requires the idiosyncratic errors to be IID across $i$ and $t$, quite a strong assumption. Also the regressors should be strictly exogenous, but it's the same for linear FE models. In your application, the fact that FE logit wouldn't converge will make a good argument against FE logit, and will satisfy some referees but not all.
An important drawback (or perhaps more like a "feature") of FE logit is that partial effects (e.g., average partial effects) are not obtainable. This is because the partial effects rely on $\alpha_i$ (fixed effects), which can't be estimated by FE logit.
If you are willing, and I strongly recommend it, please consider Correlated Random Effects (CRE) probit or logit models, where time-invariant unobservable heterogeneity correlated with $X$ is accounted for by the Chamberlain-Mundlak device. The CRE probit (see section 5 of the link) is also known as Chamberlain's random effects probit, and are very popular (certainly more popular in application than FE logit). You can get average partial effects for CRE probit models.