(1) (3) wait for @AnkerOfficial
(2) I have three ideas which are basically hacks and not ideal but probably will work.
Use a different Powercore, likely has a different lower threshold. My guess is the Powercore+ mini.
Place the USB output into a USB hub and connect another device which draws a small but sufficient amount of power to keep tricking the Powecore into draining power. There are (were) these small thumb size $1 type devices which take USB power input.
Use solar. Solar is good for servicing smallest power needs. Is you low-power device having a small internal battery to last less-sun times, or is its usage such as only-solar workable?
Use basically an inferior product which is less efficient but keeps powering out. I have one which is inefficient from another company I won't mention which is likely more likely to work for you, it will auto-power-on for the lightest of reasons (it also has a solar panel).
I know why Anker did this, it is electrically efficient. The electronics to take the 3.7V output from the internal battery, regulate it and uplift to 5V output takes some energy. In normal typical use you'd lose only about 6%-7% from this electronics but at very small mah output the electronics become proportionally more draining. I have non-Anker similar products and they'd keep their internal electronics active for the tiniest draw like a mobile phone doing its trickle-charge when at 99% charged and a lot of energy lost in the charging electronics serving the tiniest external need.
I prefer Anker's approach it suits my needs best.
OH, and Anker tends to answer questions fastest via email to email@example.com, usually a little faster than here on Facebook, tend to be slowest to reply here.