Coordinate Reference System with units in meters for central Germany?

I suggest to use EPSG:25832 ETRS89 / UTM zone 32N, because the Gauss-Krueger DHDN CRS zones are based on the old bessel ellipsoid, and you have to apply a datum shift that is not constant over the whole of Germany.

ETRS89 has no datum shift to EPSG:4326 WGS84 which you can get from GPS receivers, and all official cadastral information has beeen moved to ETRS89 by now.

You could set up a local transverse mercator projection on a center point of your area of interest as well, but that would gain only little improvement on accuracy against UTM.

By the way, EPSG:3857 has no real meters as units, they match only at the equator.

I'd use whatever is the official CRS in Germany:

For posterity's sake, some Gauss-Krueger projection like 31466, 31467, 31468 or 31469 (one per zone). Your town falls into zone 2 (ESPG:31467):