There's some secret trick to access the Service menu that I can't get to work on my ST2. But from that Service menu, the running light can be turned on (but Stromer told me that if you don't know the right Service sub menu to go into, you can put it in a setup mode where you need to know all the right values to enter, and it can be a mess without knowing what to do).
I usually just turn on the headlight (which also turns on the U-light and tailight) because I want the tailight on for cars coming from behind.
There should be settings "DL AUTONOM" and "DAYLIGHT". Set "DAYLIGHT" to on. If you want the daytime light to stay on as long as the bike is in turned on, set "DL AUTONOM" to ON. If you set it to off, I believe when you toggle the light button on the right handlebar the daytime light turns on and off with the rest of the lights on the bike.
It sounds like maybe there's a hardware issue? There's a wire coming from the light and another from the bike, and somewhere near the handlebar post, they plug in together. When I got my bike, the guy told me this connection sometimes comes loose. If that's not it, I'd maybe call the shop about it.