If stops are correlated with prior crime, and prior crime affects present crime, then the endogeneity will bias the effect of stops on crime upward. So it is plausible that your null results so far are picking up a masked effect. Anyway you cant argue convincingly that there is no effect without clearing up this endogeneity problem.
As far as I understand your problem (I'm sure that I'm missing nuances), you either need an instrumental variable for the stops, or some other identification strategy (maybe some sort of discontinuity?), or you need to lag your dependent variable. If you do the latter, you'll need to use panel GMM to avoid Nickell bias, unless there has been some generalization of those techniques to nonlinear (like negative binomial) models that I'm unaware of. Lagging the dependent variable will only solve the problem if the only "selection" into stops has to do with previous crime rates, which may not be true.
EDIT: I just noticed that your T is very large. Nickell bias will be small as T gets big, so yeah, lagging should be more or less fine.