Try to make a few simulations with different Rloads, Vout and Icc settings.
Also try to add some output capacitance (with ESRs) to see what happens with larger capacitive loads (up to 2000uF for example).
You may make the 1ms load switch pulse longer in the simulation such you see the whole event with smaller loads and larger caps, like
PULSE(0 1 0.01 10n 10n 10m 50m)
PS: also do use the 7812 instead of V6 there, of course.