I can say nothing about TNH requirements, but it is entirely typical to require drywells based upon the lot coverage. I think the clerk here mixed up some concepts in his/her explanation. Hardscapes, whether structures, driveways or patios, prevent the absorption of rain water into the gound, and cause surface water to "run off" on a path of least resistance, which in heavy rains may not be absorbed by already saturated uncovered areas. Many homes even divert the roof gutters directly into the streets. The "run-off" that reaches the streets can overwhelm storm systems, preventing the storm water from being captured. Uncaptured storm water, along with the collective run-off from you and your neighbors' houses, patios and driveways, travel a path of least resistance to low-lying areas; i.e., rivers, streams, wetlands, bays, oceans, etc. Along the way, the run-off picks up nitrates and other pollutants, and deposits them in the waterways. This is by far the largest source of water pollution around here.
My house in Southold is far from any public road, and there are no sewers or storm drains. Nevertheless, I put in seven drywells (four for the house, two for the garage, one for the pool), even though the driveway is permeable gravel. That said, it does seem odd that you might be required to install drywells after the fact. Usually, the water containment obligations are imposed incident to a new permit. I assume when you say "to sell your neighbor needs drywells" means that the patio needs a permit but was installed without one. In most towns, a "patio" that sits on the ground does not need a permit, so you also may start with the question as to whether there is a permit issue.
Six feet by four feet is a pretty small drywell, and precast perforated concrete is not "very" heavy. It gets in the yard attached to the same front-loader that will need to dig the hole. So,It should not require a lot of clearance. The question is more whether you can get a front-loader in your backyard.