Many of the newer, high quality, powerbanks these days have a low current setting for charging Bluetooth headphones, etc. They should run these fine on that setting.
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Most devices have charging currents a LOT more than 1ma. Some of my power banks that shut off at that low of current will operate at around 100ma or more to stay on. I have several Anker power banks in several models some that are newer and none stay on at very low currents. You could probably get some to work if you ramped up the current used to 50ma but very very few stay on below 15ma.
My first favorite go to power bank for using these dimmed LED lights is a single cell walmart one that I don't think they make any more it was $4.88 called Pocket Juice in neon pink and black. I've found almost no power banks with ON switches will operate at very low current.
I have a dozen different styles/models of cheap Ebay power bank kits and haven't bought any lately that support very low current but when I was buying them the cheap plastic square ones for awhile supported it and the round 2 piece white/colored slide together ones did and the metallic colored lipstick looking power banks all did. I spent a lot of time trying to find power banks that would support these low dimmable 6 LED boards and as time went on the manufacturers changed board designs several times which made it so finding one that works is tough the odds are about 10 to 1 against finding one that works (if not lower). Store bought power banks only the one I already mentioned worked no other ones work.
Simply put these dimmable boards are nice but operating them off a power bank will almost certainly finding them shutting off when you ramp down the light output you can't run them on low on most power banks. I suggest considering buying the 3 LED non dimmable USB light modules there are several different models that run from 15-30ma or so. I know if I put 2 of them on an anker power bank it stays on but only one of them it shuts off. There are other USB lighting options also. I sort of like the corded light bulbs that you can plug in and hang up somewhere. I have all sorts of USB cabling, on/off switches, splitters, extension cables, right angle adapters. I like the right angle adapters as you can put a module in one that has double sided contact and stand it on end and have a small table lamp. With a decent hub plugged into a power bank and a handful of USB lights you can set up an emergency lighting system.