As far as I know, the neighbor has to tolerate construction measures that are important. The condition is of course that the previous state is restored. I would not pay the full price under these conditions.
The property ends with the concrete wall of the garage. To dig, I would need the area from the neighbor. [Dieser hat dort bereits bepflanzt und Pflastersteine gelegt.]
Therefore, the question is whether something can be done from the inside.