I think you have a problem as none of the Liverpool registers from K to KD are listed as surviving according to information I have. Chance of finding the original log book is rather slim. I was very lucky when I purchased my 1932 Flyer to be given the original log book complete with war time petrol rationing stamps and also a list of previous owners over the years.
It is quite likely that the bike was originally sent to one or other of the Liverpool Scott dealers, A E Reynolds, or Hemmings, and if you look up your engine and frame number in the works dispatch books held by the VMCC, you will see who it was sent to, and if not Reynolds or Hemmings, it might even give the name of the original owner. Good luck !