Reliable power batttery for 5S?

It looks like most today's external batteries are too smart, registering the current requirement for 5S as negligible and shutting down the charging. I have tried unsuccessfully RAV Power Luster Mini - which I specifically got for long day runs/hikes, and Anker Powercore 10000 - for multidays outings, they both behave the same: it charges the watch for 1-2% and a couple of minutes then it gives up. And by the way, the watch dies after 12:30h from 100% charge, with phone connectivity disabled, so definitely not a day long survivor.
Can anyone recommend a power bank that actually works, and preferably charges the watch to 100%?

Thank you.