Well, if a system is rigid, it means that the volume cannot change (therefore work cannot be done), but it does not imply that Δ(PV) = 0 as the pressure can still change. The condition for Δ(PV) = 0 is for the reaction to occur isothermally (assuming an ideal gas where the number of molecules in the system stays constant).
But, what will happen if an exothermic reaction between ideal gases occurs in a rigid, adiabatic container? In this case, you are correct that dU = -Σ(µ∙dN) since w and q are zero. Because the internal energy of an ideal gas depends only on its temperature, the fact that the internal energy has changed implies that the temperature of the system has changed. So, as perhaps is expected, the system will increase its temperature.
How do we calculate the change in enthalpy? Well, that's a bit more complicated because you would have to account both for the change in chemical potential of the system as well as the Δ(PV), but it should work out to the same value as expected under the standard constant pressure, constant temperature conditions.