I was about eight when Dragnet aired, but had a Joe Friday cap gun that had silver plastic grips with his Badge 714 in them. The gun was an obvious copy of either a Model 36/37 or a M-12 Airweight as it had the flat cylinder release. It had a ramp front sight.
I read here and there that Jack Webb was presented by S&W with both an M-12 snub and an M-39 9mm auto, maybe others. Actor Raymond Burr got a M-29, but I have no idea why, as he wasn't asociated with guns much in his roles. They did appear as evidence on his, "Perry Mason" show, of course.
If the cap gun was really patterned after the gun from the show, it was a flat latch M&P Airweight or a Chief. The cap gun was just a bit larger than a Chief's Special.
I agree that the gun seen in the link above is a K-frame snub .38. LAPD did issue the Ithaca M-37 shotgun, so it was a logical item on the show.