It seems to me that the fact that the class of Mahlo cardinals below a weakly compact cardinal $\kappa$ is stationary in $\kappa$ is in fact a reflection property. Out of the many equivalent definitions of a weaklky compact cardinal, are some based on the reflection principles?
To me, the extension property seems to be very close, but it works the opposite way; for a weakly compact $\kappa$, instead of giving a smaller $V_\alpha$ where some property also holds, it gives us an elementary superstructure. Can this be seen as some sort of a reflection principle? Is there anyone who has written on this matter?