Heat involved in isenthalpic process
After further reflection I realized an initial comment I made was flawed*. To quote the Wikipedia: 


  Isenthalpic processes on an ideal gas follow isotherms, since $\mathrm dh = 0 = c_p\,\mathrm dT$.


That means that for an ideal gas isenthalpic is synonymous with isothermal, and this implies $\Delta U = 0$ and so $q = -w = P_\mathrm{ext}\Delta V$. 

For the isobaric process described in the problem, the work done by the gas is $\pu{500 J}$, and therefore the heat expected for an isothermal process is also $\pu{500 J}$.



* The source of my error is simple: for an irreversible process the heat is not equal to $\Delta H$. In fact, since the external pressure is not balanced by that of the system, $P_\mathrm{ini}\neq P_\mathrm{fin}$ for the system (the process is not strictly isobaric for the system). For a reversible isobaric process $q_p = \Delta H$.