Welcome. My name is ***** ***** would be glad to assist.
Thanks for the details, they make a difference in understanding the situation.
It may not be what you wish to hear, but I have to provide the correct instruction here for yours and anyone else s safety.
The hot and neutral going to the lights MUST be on the same circuit. This is the NEC (National Electric Code)
Code requires that they run together for this reason:
1. Most people do not understand electricity. The white wire carries the same amount of current as the black hot wire.
It returns the current back to the origin of power, which in your case is the utility transformer.
For that reason, if you turn off the breaker to those added lights, the neutral is on another circuit which is still supplying power around the home to other items.
So, any connection that is undone on those lights while the others are still powered, will be carrying current and can cause electrocution.
On installing the lights, if you run a cable form the controlled receptacle, you would parallel the lights with the receptacles on the switch and they would work fine.
This is what is done in those cases on homes with the switched receptacles. Then you can leave the receptacles switched or swap over to only the lights.
The 3 way switch has no neutral connection of course, only 3 terminals that flip flop on the power as you flip a 3 way or 4 way switch.
The other 3 way switch probably had the power and neutral in its box.
Hope that explains the situation. let me know anything unclear.