I am old enough that I remember VERY WELL the Andy Griffith series. In fact, I remember very well watching the pilot which was an episode of the Danny Thomas Show.
The series ran about 6 or seven years beginning in the Fall of 1960 with a 1960 Galaxie in the first few episodes before the 61 was available. After that, they had the most current model Ford Custom available at the time of shooting. In my opinion some of the very best episodes were shot when they were using the 64, except for the paint job, it was JUST LIKE YOURS. At that point they had been doing it long enough to be really good, but it was before Don Knots left which changed things quite alot.
I can't imagine that there would be any problem having a Mayberry Police Department Logo on the side and a Bubble Gum Machine on top. I would only turn on the Bubble Gum Machine when at the drag strip. Turning it on while on public streets would probably get you in trouble.
Mayberry police car clones have been made in pretty good numbers. I wouldn't be surprised if you could get the graphics readily. There are companies that produce graphics for police cars. Get a shot of what the one on the Andy Griffith show looked like and I'm sure you could have it made. There used to be a company called Sargen-Sowell in Arlington, Texas that supplied Fire and Police departments. They sold such graphics.
If that car were mine and I were building a Pro Street car out of it, I wouldn't hesitate one second in making it suitable for Andy.