Use scale from an object in comp to scale a separate object in another precomp

I'm not sure that I am following you at all. It sounds like you want a background comp containing a shape layer that isn't rendered that can be used as a framing guide for rendering comps that contain the background and maybe some other graphics. Did I get that right? Your master Background Comp will have an image and 3 guide layers that are shape layers, that you can use to position the background in a standard HD comp, Vertical HD comp, and a Small Landscape Comp.

If that is right then you have to calculate the difference between the scale of the guide layer, the rotation of the guide layer, and the position of the guide layer using Position Scale and Rotation of the Guide layer in the appropriate rendering comp by offsetting the position rotation and scale so the framing in the rendering comps matches the framing of the guide layers. If that is right my comp structure flow chart would look like this:

This is what the expressions would look like:

// Anchor Point
bgComp = comp("Background Master Comp");
guidePos = bgComp.layer("Standard HD Guide").transform.position;
guideOfst = [bgComp.width, bgComp.height] / 2;
value + guidePos - guideOfst

//Scale
guideS = comp("Background Master Comp").layer("Standard HD Guide").transform.scale * .01;
[value[0] / guideS[0], value[1] / guideS[1]]

//Rotation
- comp("Background Master Comp").layer("Standard HD Guide").transform.rotation

 Set up a shape layer guide the same size as your Rendering Comps, make sure the Shape Layer Rectangle properties are all zeroed out so the rectangle is centered on the anchor point, and change the Layer name to match the guide and positioning the guides will position the background. 

I would also add these expressions to the shape layers contents properties so that you couldn't inadvertently get things out of alignment. Just modify the comp names and set the comp sized guide layers up like this:

Here's a sample comp for you to play with. Use your own footage and fiddle with the comp size of the Background Master Comp. I made a short video showing how the project works.

If that is not what you are trying to do let us know. It's not how I would have handled the project. I would be using Master Properties.

I'm not going much into detail here, but things should get easier if your shapes original size is equal the size of your picker comp. A scale of 100 is something you'll probably never use, but the math will get way simplier, just like you did on rotation.

*Martin