Exactly what part of the septic system is on the neighbor's property?
When our drain field had to be repaired, replaced, and raised, due to the environmental protection needs, we were limited to where the drain field could be placed ON OUR PROPERTY. The environmental health inspector and the entire field team worked with the builder and the septic tank company to keep it legal and least expensive to the builder and to us.
If the actual septic tank is on the neighbor's property, then you have a cost of new septic tank, and new drain lines for consideration. If it is just the drain lines, then a new set of drain lines can be prepared on your property.
In our case, we ended up having four inches to the property line and raising the drain field (which meant adding windmill pipe arrangement to the drain line), to give us the necessary requirements for the septic tank.
The original septic tank was cracked and was repaired while the ground was open, the new holding tank was put in where the old drain field was, and the new drain field ran around the side of the house because to put it in the back would be too close to the wetlands.
Nothing can be removed from the property unless it is treated as biohazardous waste (which makes it expensive). We spread out the dirt across the inside of the property line, put lime on it to defuse the smell immediately, and the neighbor was unhappy with the view of the drain hill. The next part was landscaping the area in such a way to not obstruct the drain field, so in front of the hill, a quick growing shrub group was placed, which hid the hill.
Best wishes.