The technique for getting to ~3.1km/s is just from having done lunar missions far too many times!
The adjusting prograde technique is fine. However, because the projection drawn to the screen is normally ecliptic, and the ISS's orbit is highly inclined to this orbit, it will seem as if the orbit is not as eccentric as it really is.
Simplified explanation follows.
If you do your TLI burn when you are on the ecliptic, the apogee will be similarly on the ecliptic - which happens to be in the orbital plane of the moon*. This is where you can use the 'adjust-prograde-until-it-brushes-the-orbit-of-the-moon' technique. However, if you do the TLI burn at a high latitude, then your apogee will also be at a low latitude. Since the MFD is drawn from above the ecliptic, even though the apogee will be the same distance from the earth as the moon, the orbit will seem as though it its apogee is well inside the orbit of the moon.
You can see this change as you advance the menouvre date in hyper. The apogee of the orbit seems to grow and shrink wildly as you move the time of the burn around the orbit, reaching its maximum when the point of burn is on one of the nodes (points where two orbital planes intersect - denoted by the grey line).
The technique would be similar to how you've done it before with increasing the prograde until it grazes (or is slightly bigger than) the orbit of the moon - except in this case, set some DeltaV so that you get a noticably elliptic orbit. Then adjust the Menouvre time so that the Apogee/Perigee of your hypothetical orbit are on the node (grey line). Then adjust the Prograde DeltaV so that it's grazing the orbit of the moon.
*Actually, the moon's orbit is inclined at 5° to the ecliptic, but ignore that here.