Sounds like its probably a powerbank issue.
Using a usb tester, the 3ds only draws around 400mAh-470mAh when charging.
Lots of powerbanks turn off when they detect little load. Your powerbank may turn off when there's less than a certain amount of power being used.
Try different power banks (head to the dollar store and see if those ones work, if they do, you've only wasted $1 figuring this out)