As far as cavity shielding goes, copper foil is best for an instrument with vintage value as it can be peeled off later with no change to the instrument.
It also gives a more perfect shield, but two good coats of shielding paint gives a more than adequate shield.
For an LP Axcess I'd recommend the paint as it won't hurt the resale and it is easier to do a good job with it.
You can get small cans of the paint for about $12 on Ebay - enough to do two guitars.
You should shield every cavity where the signal is present, so this means the pickup cavities, the control cavity, and the switch cavity. You do not have to do the trem springs cutout.
Once each cavity is shielded you must tie the cavities together. I use #4 wood screws and #4 lug washers with small gauge wire running from cavity to cavity. Shielded paint which is not tied to ground does nothing. So get all the cavities tied together and then have this system tied to the signal neg (ground) at only one place.
Note: you only have to do the screws/lugs/wires from each pickup cavity to the control cavity. The pickup selector switch will tie the switch cavity to the system shield and also shield the wires going to/from the switch.
For every distance traveled through an unshielded area (the rout from switch to controls, the rout from pickup cavities to controls) you must use shielded wire otherwise you undo all the shielding.
Run a wire from the trem claw to the lug in the control cavity - this ties the strings to the system shield.
Just to be clear, inside the control cavity will be a small screw going into the side of the cavity wall. This screw will hold a small lug washer against the shielded paint. There will be wires going from this lug to matching lugs/screws in each pickup cavity, a wire from this lug to the trem claw, and one wire from this lug to the signal negative, typically at lug 3 of one of the volume pots.
Each pot case will be tied to system shield through the shielding paint.