In the Sannow book, it mentions that the reason for the use of Belvederes, rather than the full-size cars back then, was due to the width of some of the LA area freeways. Too narrow for the big cars, but the intermediate Belvederes did just fine.
Many feel the Ford Flathead V-8 was "the hot thing" back then, but the stated power was not that much more than a similar Chevy or Plymouth flathead 6! In some cases, it could come down to who could drive best. There were special intake manifolds (multi-carb) and "split exhaust manifolds" for them too.
The moonshiner cars had reinforced rear suspensions to carry their loads and not ride lower in the back. Truck tires were "race tires", as they were stiffer in construction than normal passenger car tires. Probably some sort of higher-perf brake linings, too.
When the horsepower race of the middle '50s began, many of the more powerful engines were termed either "Power Pak" or "Interceptor V-8" (Interceptor being the biggest dog in the hunt), as the Ford 352s were. If a law enforcement agency bought "the big engine", it was dubbed "Interceptor" officially or unofficially. In the case of the '58 Fords, the Cruise-O-Matic behind the "Interceptoer 352" was officially "Interceptor Cruise-O-Matic" (on the shift quadrant lens). In the earlier '60s, Ford did have a "law enforcement only" 390 engine. 330 horsepower. solid lifters, hotter cam, etc. Very similar to the earlier '60 352 360horse engine. All of this to compete with the Chrysler 383s and 413s. Chevy had their 348s and 409s, but they were in the minority, as I remember.
The CHP had wheelbase specs, which generally ended up being Mercuries, Buicks, Olds', and Chryslers/Dodges. The Dodge Custom 88s was born as allegedly the CHP wanted a Chrysler product in the 122" wheelbase spec, but didn't want to purchase "Chryslers". Read the police car testing articles in
www.allpar.com.
If the OEMs didn't supply things like high output generators/alternators, there would be aftermarket companies that did. Installation of those things were a part of the bid specs. As the police car business was more "cherished", the OEMs started doing it themselves in "Police Special" models.