This has been a long time in coming being as I've been too busy making them for comission to do pictures, but finally it is here and you can all rejoice.
To build an LpL CS, you will need a few things, and it is much cheaper to do so if you are also building a +Bow
You can find the details of the materials needed in the L+L CS SDK Thread this guide will cover construction of the LpL CS.
Tools Needed:
Scroll Saw or JigSaw
Drill Press or handheld Power Drill
Phillips Screw Driver
Knife (To clean the edges of the polycarbonate)
6-32 Tapping Bit
13/64" Drill Bit
5/32" Drill Bit
7/64" Drill Bit
5/16" Spade Bit
3/8" Spade Bit
9/16" Spade Bit
You will also need some full sheet label paper to print out the templates on.
Print out the templates and cut them out, lay them out on the polycarbonate along with your +Bow templates, it should only take one sheet of 1/8" and one sheet of 1/4" total.
Cut out the polycarbonate pieces using a scroll saw or jig saw, and make sure to cut out or drill out all the interior holes as well. Pay special attention to the catch drilling because you are going to use a 5/16" Spade Bit, NOT the 9/16" that the template indicates. If you lay the spades out you'll see that one fits the hole marked and one doesn't.
Cut the main spring in half, leaving a 6" piece, you can cut it to 6.5" if you want but it's not going to help anything.
Cut a 6.5" piece of PETG tube, yes this wastes 4.5" of a 12" piece, but I'm sure you can figure out something else to do with it.

This is what you should have when you're done.
1 Set of Polycarbonate pieces, cut and drilled
1 Main Spring, cut to 6"
1 Catch Spring (You'll have to find one of your own, the one in the pic is an Ace, I think it's #178)
1 6.5" length of PETG Tube
2 3/4" Lengths of 1/2" Nylon Rod drilled through the center with a 5/32" Bit
3 3/4" F to F standoffs (Yes, there's 5 in the picture, you need 3)
4 1/4" F to M standoffs
2 3/8" Unthreadeded Round Spacers
1 1/2" OD, 1/8" Length Unthreaded Spacer
1 1/2" Screw Size 1-3/8" OD Extra Thick Reinforced Rubber Washer
8 Nylon 6/6 Flat Washers
5 3/4" Screws
1 1-1/4" Screw
2 2" Screws
9 3/8" Screws
10 1/4" Screws
3 Lockwasher Integrated Nuts
1 SCH40 PVC Pipe Fitting 1" to 1/2" Compression Fitting

Use your Tapping bit to thread all the appropriate holes, two on each trigger housing piece, two on each grip piece, one on the catch plate and four on the fixed catch plate.

Peel off the templates and the protective plastic from all your polycarb pieces, leaving you with your materials.

Assemble the catch, gather the two pieces of the catch together and place 4 nylon washers over the holes on the fixed catch plate, put the other 4 nylon washers on 4 3/4" Screws. Place the catch plate over the fixed catch plate and insert the four screws into their corresponding holes. Tighten them down almost all the way, making sure that the catch plate can freely move vertically but that it doesn't rattle back and forth.

Gather the pieces pictured above in order to assemble the grip.

Screw the M to F standoffs into the threaded holes on top of the trigger housing pieces and use3/8" screws to attach the trigger housing pieces to the grips as pictured.

Use the 3/4" standoffs to attach the two resulting pieces together, two standoffs in the grip and one to attach the lower part of the trigger housing. Use 1/4" screws to attach.

Your completed grip should look like this.