The lowest light levels appear still to be high in your dark room picture. The only way I have found to get down to very low levels (<30 counts), is to put the sensor in a black box, or bag, and also turn the room lights out. The sensor is extremely sensitive. Try and get your background counts lower, and let's see if that side peak disappears.
But, that peak could still be a light leak/reflection or something odd happening to the sensor itself due to light saturation. The light sensor is a made up of a lot of analog transistors, and they react in funny ways when you overdrive them.
It is not a deal killer to have a single pixel be high, especially one on the edge. I could just modify the firmware to not to look for peaks at the edges or you could get a new sensor, but let's check everything first. Also, I would rather you continue to set everything up with the current sensor and then when you are done, get a new sensor.
I checked your first picture you sent, and the light profile did not have that side peak.