DVLA won't work if its an early K2 as it would have had a green old style log book and they did not keep copies of those, only the V5 and V5C
Not necessarily so, I applied for the full history for my K0 (reg'd Jan 1970) and the earliest document they sent back was a photocopy of the old style folding green log book. Admittedly it's a dreadful copy, but it is legible.......just.
If it's a UK bike the worst that can happen is you waste £5 if you don't get the info you need, but for that money and the small effort involved it's got to be worth asking surely?
Mick........kettle738