The BEC gets hot, resistance increases, voltage sags, BEC switches off, servos stop working. Brown out. I assumed you were running LEDs off 5v rail.
I haven’t used esc’s with accessory cables, I imagine those just draw straight off the battery, in which case the connector current will likely be the limit. There’s several types of JST, the ones found on receiver batteries are 10A max. If you have one of those then you are fine for quite a lot of lights.