Finally, the bike is back to life (hope the joy will last for a long time now).
Now, about the issue.
The eternal problem with the rear cylinder not switching on after riding and stoping and starting the engine. It would switch on after about a minute though. Nothing helped: almost every sensor had been changed. S&S suggested to change the whole VFI module. That was going to be the last step before changing the crank position sensor. While waiting for it from Big Dog I opened the battery tray again to check the wire connection... And here it is, laying on the bottom of the frame, covered in dust, was the cause of all evil, the relay. It simply fell off some day, causing all the trouble. I am not able to upload a video, but it was shaking badly in its connection. The only solution so far was to fix it with a cable tie. In future I am planning to change the wiring and, maybe, downgrading it to a carb to get rid of the mess. To sum up, if anyone is thinking of buying a VFI Big Dog, go, do something nice, have fun, enjoy life and get this shit out of your head. To Big Dog guys: fire you electricians. I don't know which influence they are under when assembling your bikes: booze, drugs, mushrooms or any other shit, but unless you do something about the electrical part, no one is gonna buy your new bikes, at least not overseas.