It is unlawful to depart the US without a valid US passport, but this is almost never enforced, and there is no penalty. I have never tried leaving without my US passport, but I usually don't show it to anyone when I leave, and I've never been prevented from leaving the US or asked to show my US passport or to prove my immigration status. However, I suspect that some US airlines might do so; I usually fly on European carriers.
There is a small chance that you might be spot-checked by CBP before leaving, and they might stop you from leaving. The risk is small, but if you want to eliminate it altogether, you can make an expedited passport application. It takes just a few days.