The powerbank I do trust as the sx260hs fires right up using the very same cable.
While it's not too likely, the circuitry in the S110 might be a bit different and shut down on the 10% over voltage from the powerbank.
The fake battery contacts might be a tiny tad lower (sunken into the battery) than those of an actual NB5L battery. I could use a small amount of solder to 'fix' that, but would that be the right approach?
I'm not a big fan of using solder that way as the surface tends to tarnish quickly, leading to a higher resistance contact. But it might work to tell you if that's the problem. Another approach might be to gently bend the contacts up a bit (if that is possible).
Pressing onto the fake battery with the battery door open does not help.
The camera will not start with the battery door open (unless you also press the secret battery door switch - harder to do on some cameras than others). There is a CHDK hack that will allow the camera to start with the battery door open. Unfortunately it was decided not to make this a standard CHDK feature some time ago so you'd need a custom built to use it. In any case, if the battery top contacts the inside of the battery slot before the contacts engage then pressing harder won't help anyway.