MLE or instrumental variables

I'm trying to estimate a model in which one of the explanatory variables is correlated with the error term. As I see it there are two alternatives, specify the likelihood function and maximize it to get the estimates of the coefficients, or use instrumental variables.

There are pros and cons to each approach, so I'd like to get some other opinions on the matter. the IV method is quicker, since it works with closed form equations, but the drawback is that I'll get different estimates for different instruments, which I don't really like.

Any advice would be welcome.