What is the most reliable 6S 18650 balance protection board ?

You don't have a lithium charger. The board only provides protection, it doesn't manage charging. And the power adapter is just a voltage-regulated power brick. What you need is a 6S charger; here's one I found:
https://www.ebay.com/itm/324355305254

As for why it would cut out under load; maybe you don't have it connected correctly? Maybe it's defective?
And, 15 amps is the current through all the batteries since they are in series.