Exceeding the floor area ratio

Below again the image of the textual determination.

Almost unreadable.

only the reference to paragraph 19 of the federal land utilization ordinance.

That's great after all.

Do I now have to submit an application to the municipality,

No, it is exempt from procedural requirements. Nevertheless, the following applies (as already stated):

"Also, construction measures exempt from procedural requirements must comply with the other provisions of public building law. In particular, the projects must not violate urban planning law. For example, the location of a tool shed on your property or the construction of a fence can be restricted or excluded by the provisions of a development plan. Therefore, the regulations of a development plan or the rules for building in the outer area must be taken into account when constructing buildings exempt from procedural requirements. Furthermore, the boundary distances prescribed by the Lower Saxony Building Code must be observed."

I'll put it this way: if someone complains, then you should be able to produce an exact and comprehensible calculation.
By the way, it is worthwhile to skim through the federal land utilization ordinance, because one paragraph refers to another.

Thank you for the feedback so far. In fact, I have also complied with the other regulations until now. I would prefer to build the garden house about 2 meters away from the boundary. At the boundary, there is a hedge in that spot. Otherwise, there is a garage 9 meters along the mentioned boundary. Therefore, unfortunately, I have to keep a 3-meter distance with the garden house. Are there any other possibilities? Unfortunately, I am not getting any signatures from the neighbor. I probably wouldn't do that either.

Therefore, I unfortunately have to keep a distance of 3 meters with the garden house.

That's how it is.

Are there any other options?

No. Not known to me. I am also from Lower Saxony.