Manipulate path points as a group in a precomp?

Directly, there's no way to do this with points on a motion path. There are less-direct methods that might work for what it looks like you're trying to do here, though. 

Use the pen tool to draw out a path that you want your layer to follow, creating a shape layer. You can disable the visible Stroke and just use this layer as a holder for this path, though the visible Stroke can make it easier while working. If you then select multiple points on this path, you can double-click any of them to transform them all in the manner you're requesting. 

To avoid rebuilding what you've already built, you could just copy that existing motion path (i.e. copy the Position property), and paste it directly into the Path property of this newly-created shape layer. It's possible you'll need to re-position the entire layer, but it'll bring your path over intact.

Now, you can't link your layer directly to this separate path, but you CAN link it easily. 
Right-click on Path Property > Keyframe Assistant > Null Tracing Along Path

This creates a Null Object that traces along the path, and keyframes for you to control the progress along this path. This is aware of any changes made to the path, even if it animates over time. Simply remove the Position keyframes from your visible layer and parent it to this new Null. (Shift-Parent to snap it to precisely the Null's Position). 

While probably overkill for your example, you could also connect each path point to its own Null for even more control (and the ability to parent those together, or to additional nulls) - See Null Controllers for Path Points within that same submenu referenced above.