Here's an update:

I went to the Cheguansuo in Hamilu today. Unfortunately, it is only possible to do this kind of buerocratic stuff during the work hours Monday to Friday, 9am to 5pm. So I had to wait until our company shuts down due to power blackout, which did not take that long.

After taking the pictures, doing the 体检, paying several small bills and waiting for two hours in building #1 the unfriendly officer (why are they all dick heads? Ok, the girls at the 体检 weren't.) told me I should go to Beijing first and change the passport number on my driving license. Then I could come back and change the address in Shanghai...
Gee, what an uncomplicated process. Why have access to other cities' system in the same country, if you have some annual vacation left, can buy cheap flight tickets and just fly there?! But for the really lazy people, who prefer to use their vacation for family time and their money for PM2.5 masks we still have agencies, who do their guanxi-magic for just half the price of the air fare.
Sometimes I hate this country... ok, most of the time.

So that's the next step: tomorrow i will shunfeng my old and new passports together with my Beijing DL, an original Paichusuo Dengji Biao (police registration) and a one pager description of the situation (情况说明) in my best Putonghua to a trustful agent in Beijing I just googled. He claims he can do it and shunfeng everything back in two days for just 800 precious RMB. (They have an English website, so that's enough for me to trust him with my passports and residence permit.)

Ahh, China, what a lovely country. Too bad they don't sell firearms here. I could have used some today.