It's been my experience with 8.8's that the shims are related to the housing and not so much the gears.
IOW, most of the time using the same shims that came with the rear axle works as long as you're using FORD gears. Aftermarkets are another matter.
In this case it would mean that you would need to remove the pinion shim from the pinion coming out of the aluminum case and transfer it to the pinion you are taking from the iron case. You may find they are the same but you can't count on it.
Carrier shims you'll be removing anyway, so no difference there.
Since you will be removing the pinion gear, the pre-load on the pinion bearings will need to be reset. Preload is merely how tight the bearings are from how much the pinion nut is tightened. Experienced techs will flat-rate the job and just hammer the pinion nut down with an impact until it feels right to them. There is a spec and it requires a inch-pound toque wrench and you tighten the pinion nut until it takes the specified torque on the inch pound wrench to rotate the pinion gear.
Between the amount of torque it takes to run the pinion nut down (it's an interference nut) along with the amount of torque it takes to crush the crush sleeve we're talking about a serious amount of force required, sometimes more than what a quality 1/2 inch drive impact can do. Add in the need to press on and off bearings along with a few more details, and I'd suggest finding someone to do it. Try finding someone in the local late-model Mustang crowd and they can probably point to someone who does that stuff on the side.
edit
Is it possible to pull the limited slip clutch set and springs and install them in the "standard" differential?
Just saw this, and the answer is No. You'll have to change the carrier. The one out of the iron case will
probably fit but I don't know that for a fact, not having dealt with an aluminum housing before. They're both 8.8 designs, though.