I am also no expert on Lugers. But I think they are cool, futuristic looking, and downright sexy.
In 1976 my mother gave me a 1918 DWM for my 21st birthday. It was stolen in 1985 and I found another 1918 DWM to replace it (like that could be done). This one was stamped for the Wolfsberg Police Department. As I understand it there is some sort of safety modification made to the police guns to prevent them from firing during disassembly. Since you are a collector of police weapons you might want to seek one of those. Sellers will usually mention this feature in the ads.
The difference in price between a mismatched shooter and more complete and original shooter isn't all that much. I would look for a nice (but not pristine) example with the police safety and often they have departmental stampings like mine and that should appeal to you.
Mine is all matching and in pretty decent shape. Very reliable (Lugers can malfunction with low powered ammo as they were designed for full power military loads) and pretty accurate. I used it in a service pistol match about 20 years ago for fun.