I've been checking all the questions and can find many similar ones, but unfortunately none that addresses my problem. I ended up reading many interesting questions and answers (including e.g. this one Can i connect Power source (like usb charger) parallel connection with power bank output for keep Uninterruptible power supply (UPS))
I however already have a few PowerBanks mainly something like this. I would like to use one of them as a ups, but my problem is it does not seem to charge when it is charging. Is that a "feature" of the battery, or is it to "protect" me from wearing the battery down? How difficult/dangerous would it be to disable that feature? Could i just go ahead and get the power directly from the end-poles of the batteries bypassing all circuitry, or that would be dangerous?
Last, I have a second brand of powerbank, this one. Here things are even more complecated, as the battery requires me to press a button for it to start charging… Is this standard?
I have seen it be done, since I have found that there are similar power banks out there that can fact provide while charging e.g., with other power banks, e.g. http://raspi-ups.appspot.com/en/index.jsp. It also seems to be an "ok"-ish thing to do (https://www.quora.com/Is-it-okay-to-charge-your-phone-through-a-power-bank-while-the-power-bank-itself-is-being-charged)
Power banks are a class of devices to serve as emergency power sources in stand-alone mode. You charge it, and keep in your briefcase just in case. As such, most power banks are not designed as UPS (uninterruptible) supply. While some banks can do this by design (as the OP has discovered this himself), it does not mean that all power banks would charge its internal battery and provide power to output port. Some power banks even provide a warning label, saying "Do Not Charge & Discharge At Same Time!!!"
The powerbank can not be modified as UPS without replacing its active electronic circuit. The new circuit must contain the charger for battery with direct path to VBUS, AND an additional automatic power switch from direct feed to the battery booster to OUTPUT if external power is lost. Texas Instruments has a sizable portfolio of PMICs (Power management ICs) where this function is integrated into controller IC, as BQ24075 for example. Other manufacturers as Linear Technology, Maxim Integrated, Richtek and others have similar ICs. This board from SparkFun can likely do the job.
ADDITION: Here is a solution for 1-cell UPS offered by Maxim.