Passing the integer district ID will return only the requested representation of the district. The geometry is converted to GeoJSON format using shapely and the to_shape method from geoalchemy2.shape:

@app.route('/nba/api/v0.1/district/<int:district_id>', methods=['GET'])
def get_district(district_id):
  district = session.query(District).get(district_id)
  shp = to_shape(district.geom)
  geojson = shapely.geometry.geo.mapping(shp)
  data = [{"type": "Feature",
  "properties":{"district":district.district,"id":district.id}, 
  "geometry":{"type":"MultiPolygon", 
  "coordinates":[geojson['coordinates']]},
  }]
  return jsonify({"type": "FeatureCollection","features":data})