Is the neighbour open to idea of the problem being addressed, albeit not at his own expense, ie would he have a problem if you did the necessary works, obviously requiring access onto his property and some local disturbance to his driveway whilst appropriate works are carried out?
With all you've probably spent so far this approach would have been more appropriate.
ah hindsight!
the problem is there is no evidence that the neighbour's drive categorically "is" the cause of the problem, after all, the surveyor said he couldn't say for sure.
The neighbour may be open to my paying to have his drive dug up and lowered but, and I might be being pedantic here but, that's an insane idea. If his drive is the problem, why should I pay (A large sum of money) to have it corrected, -not him? If his drive is not the problem what good would paying to get it lowered do?