Adding lights to a 3-way switched circuit
Smart switches

Use smart switches for this.  This makes the impossible easy. 

Traditionally, you would need one cable with 4-5 active conductors plus ground. And all those must be in the same cable!! You cannot group two cables as a substitute for a cable with enough wires, ever. Nor  can you use any #14 wire on a 20A circuit.

Here, we re-task white to be neutral, black to be always-hot, red to be switched-hot, and obtain a matching pair of master and remote 3-way smart switches that communicate wirelessly or via power line signaling. 

Given the ease of this approach, doing traditional 3-ways is a waste of effort.