It is hardly surprising that it fails to converge. As you have shown, the response variable is highly zero-inflated. Even if it did converge, I would treat the results with great suspicion.

You need to use a package which allows for a zero-inflated negative binomial response, such as MCMCglmm.

It is hardly surprising that it fails to converge. As you have shown, the response variable is highly zero-inflated. Even if it did converge, I would treat the results with great suspicion.

You need to use a package which allows for a zero-inflated negative binomial or zero-inflated overdispersed Poisson response, such as glmmADMB, glmmTMB and MCMCglmm.