Okay, I've tried every suggestion here without success.
Washers: I couldn't find any teflon washers, but I tried regular ones and plastic washers (dipped in vaseline). I can't put washers on both sides as there's not enough room, but I tried both ends and it didn't change a thing. When I put a washer on the lower side it is just hanging in there, not touching anything, so we can skip this washer imho. Hopefully I can find some teflon washers somewhere and I really wish that they'll do the trick.
Turning the bridge 180°: I'm sure the nuts aren't touching the pickup, but I tried this as well. Flipped it over and the issue remains. But thanks for the advice and maybe I'll keep it that way, because it seems more accessible from this side.
The non rolling roller: tried as you suggested, put a string below the roller and tried to use it like a v-belt. It's stuck. Seems like the bracket has been pushed together too close and the roller is stuck. Will try to bend it open gently.
Finally -> Lube: I applied some vaseline on all moving parts involved. I even put a small amount on the zero fret and the head's check rail (is this the correct word?).
Still, when I push the arm down it comes back up to the right spot and the guitar stays in tune. When I pull it up, it seems to get stuck a whole note above the original tone. I wiggle the whammy a lot and this is causing serious trouble, as I can never tell if it falls into the right position...
I will order another spring asap. It's my last hope (apart from the teflon washers). If that won't work I'll have to keep an eye out for a complete new system.
Or maybe buy a Hallmark Whammy.
One last thing I noticed: when I flipped over the bridge I had a hard time getting it out and putting it back in place. Seems like the bolts that hold it in place are too close to each other (but just a little, not much). I have to be careful to put the bridge on both bolts at the same time, otherwise it gets wedged. But the bridge seems to float like it should...
Any more suggestions? Can I use a regular Bigsby spring as replacement? It will be hard to find an original replacement over here.
Thanks so far to everyone who replied! This kept me going. Even though I haven't been successful by now, it feels good to have an idea what you could do about the issue. I really appreciate your help! Thanks!