OK, here is what happened today:
I thought I haven't got much to loose to try anyway. After a few minutes I was called into the main building, then the director of immigration took me into his office where he explained to me that according to an agreement between the two govts. only cars, trucks and buses are allowed onto the bridge. He then proceeded to call his counterpart in Thailand and they agreed to exceptionally let me cross.
After that processing was quick. Nobody asked for any bike documentation, which I didn't have anyway, only had to show the rego.
Approaching the toll booth I was stopped again. They had a problem with me crossing and wanted me to go to the police, which I didn't, obviously. I think their problem was that the toll system is automated and they don't have a rate for bikes, but there are induction loops in the road. After 10 or 15 mins they indicated that I could go, but I had to leave via the wrong way, where there was no barrier or loop.
So, I made it in the end, but I would advise against others trying it, because if it doesn't work you have a long detour ahead.