Linear Probability Model Instead of Logit in Fixed Effects Regression
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.