First that is a very well detailed, precise question.
Okay the problem is that you mentioned heavy. I wont be able to fulfill that requirement entirely. See the gluteus medius isn't dynamically involved in hip extension which is the function more associated with big compound lifts (RDL, goodmorning, squats, reverse hyper, back extension). At best it is used as a fixator during big lifts which are done with a wide stance, but as a fixator it is only used to maintain proper position so that the other muscles can do the lifting.
The main function of the gluteus medius is to move the leg laterally away from the body (either through abduction or transverse adduction). A such movements that will focus on the gluteus medius are the exercise where you move the leg out to the side (think standing with a pulley attachment fixed on your ankle and moving the leg outside and away or the abduction machine where you spread your legs open).
I know these aren't really in the realm of "cool" or "heavy" exercises but they are the ones that will hit the muscle you want to develop.
I found the best variation to be to use the abduction machine and do the exercise leaning forward a much as possible, this almost cramps the gluteus medius.
One "multi-joint" exercise that can work (and I wish I had it on video since it's kinda hard to explain) is the 30 degrees split squat with added band resistance.
I will try to explain it, and worst case scenario I'll will film it tomorrow.
Attach a resistance band to a post (can be the post of a power rack).
Loop the band around the leg that will be the back leg in the exercise (as close to the top as comfortable).
Set-up in a split squat position in such a way that the band is trying to pull you toward the rack and back (you want a fair amount of resistance).
The feet position should be a bit wider than a regular split squat. I do not mean that the front foot should be further forward, but that it would be further to the side as a in a regular split squat.
Do the split squat but focus on pushing with the back leg. In a regular split squat I want people to go straight down and keep a 50/50 weight distribution on both feet. With this variation I want you to push a bit forward and out, as if you were trying to push your body toward the front foot.
When you are in the low position the back leg should be a bit extended because you pushed your body away. The band will add resistance to that pushing action and will recruit the gluteus medius.
Hold the low position for 2-3 seconds, focusing on flexing the gluteus medius... you do that by pushing isometrically with the outer edge of the back foot.
Then slowly return to the starting position.
And no you can't simply do lateral lunges or lateral split squat because the abduction function is not loaded in there movements You need the band to load that portion of the exercise.
At first I would probably only use band resistance (using jumpstretch/elitefts resistance bands) and try to increase band resistance (either by using tougher bands or standing further away from the rack). Adding dumbbells will only make you load other muscles groups and it will be harder to focus on using the gluteus medius. As you become effective at recruiting it you can add DB resistance.