When you have two switches in this configuration, you have to look at one important detail. Power is supplied to one switch, and the switched power comes off of the other switch and goes to the lights. In your diagram, "from source" wire leads to the switch on the left. The switch on the right has a black wire coming from it that supplies power to the lights.
Because of this, the only way to add another light is to get power from one of the existing lights. You can't pull power from the switch on the left because it doesn't have a wire for switched power. You can't get power from the switch on the right because it doesn't have a neutral. The only two boxes that have switched power and neutral are the two light fixture boxes.
Traditional "steam" wiring
Nope. Cannot get there from here. What you want is simply impossible. The wires you need are simply not present in the places you need them.
Your option is to abate the asbestos, replace the inter-light cable with a 12/4 cable, and extend off one of the lights to the additional.
Naturally, this is easy, and can solve the illegal inter-lamp wiring as well You need a smart-switch "master" and a smart-switch "remote". Both can need AC power but they must be able to communicate via powerline signaling or wireless (e.g. Insteon).
- On the lower 2-wire cable, we use colored tape to re-mark the black wires red on both ends. It's already switched-hot, but now we're marking it. Also mark the black lamp wires red, since you want them hot when switched, yes?
- On the right side /3 cable, the red wire is capped off. It will not be used at all.
- In the right side lamp box, the right side /3 cable's black and white wire go to the upper /2 cable's black and white wire. No color changes.
- Important: again in the right side lamp box, the lower and upper /2 cables do not connect in any way. They stay 100% separate. (their ground wires can connect).
- Almost there. In the upper LEFT lamp box, all whites connect (both /2 cables, the /3 supply, and the lamp). The two blacks connect (/3 and upper /2). All 3 reds connect (/3, lower /2, and lamp itself), remember we remarked 2 of those red!
- In the right switch box, wire the smart switch "remote" to always-hot and neutral. Red is unused.
- In the left switch box, wire the smart switch "master" to always-hot (line/black), switched-hot (load/red) and neutral.
- Now the 3-way smart switches should drive the old lamps.
- In the left switch box, mark the black in your new cable red (it's switched-hot) and add it to the other red.
- Join the new cable's neutral to the bundle of neutrals.