As you probably know, Nosha, there is a Dispute Resolutions Procedure defined as part of the Act. Its primary purpose is when a neighbour won't cooperate in the first place - not towards the end of the job.
The way it works is that an 'Agreed Surveyor' gets appointed who assesses the job and defines what should be done. You, as the guy wanting the job done has to pick up the tab for this, but the individual must be an independent professional.
Whether you can now claim the Engineer you commissioned was the 'Agreed Surveyor' I don't know, but I would be inclined to go down that route. That would imply writing to the neighbour stating that you have an a 'Surveyor' assess the situation and with recommended works, and that is your position on the matter. Then do nothing.
Otherwise you are going to have to get involved in more costs of Surveyors - perhaps one appointed to them - all at your tab cost, for a few cracks that might cost a day's worth of decorator time to put right