It's been a long time and things may have changed, but I was able to obtain a police cert through an agency in 2015. They were able to do this for two different periods for me - the first I stayed on campus at a uni and had no household registration, the second I lived in a flat and had the proper registration. The only information the agency required to obtain the certificate was pictures of entry / exit stamps in my passport, picture of my passport and a power of attorney giving the agency the right to act on my behalf. Has the law changed since...? They turned it around in a month for me and the entire process was seamless. Hope you can get this sorted, it sounds frustrating!