To the OP...
FWIW if the gun has an adjustable rear sight, it may be a Regulation Police Target Model, which are rare and are worth about 3x standard value.
You are correct that these guns were discontinued for a while. The .38 Regulation Police was made from 1917 until 1941, when S&W discontinued production of all revolvers on the I and N frames to concentrate on K frame guns for the US and British military. The British military actually received a few I frame .38 Regulation Police guns early in the war but evidently decided that they were not suitable for use as a service handgun.
IIRC the gun was reintroduced in 1948. In 1957, it was given a model number along with every other S&W product, becoming the Model 32. It was discontinued in 1972 along with all of S&W's other revolvers chambered in .38S&W, the .38/32 Terrier (Model 33) and .38/200 Military & Police (Model 11).
The real serial number on a Smith & Wesson is on the bottom of the butt of the gun. It might be covered by the grips. On some older guns it is also on the bottom of the barrel under the extractor rod and on the rear face of the cylinder. The number under the crane is an assembly number used only for tracking inside the plant.
Prewar Regulation Police guns have a rebated backstrap and extension stocks that cover the bottom half of the backstrap and the butt. This gun is
one of the exceptions to the rule that S&W serial numbers are on the butt; the serial number of a Regulation Police is normally located on the frontstrap beneath the trigger guard.
Prewar S&Ws that were normally sold with extension stocks usually have the serial number on the frontstrap; the same goes for prewar .22/32 Target Models aka Bekearts.
Except for a handful of early-production guns made from stockpiled prewar parts, postwar Regulation Police revolvers were built on a conventional square-butt frame without the special extension stocks. They have the serial number in the normal location on the butt.
You are correct, however, that a prewar gun should also have the serial number repeated on the barrel flat and the cylinder. On some prewar S&Ws, the cylinder number is under the ejector star, but IIRC it should be on the cylinder face of an I frame. You are also correct that the "345" inside the crane is a meaningless assembly number.
Dated grips are getting you into interesting S&W territory I am not qualified on.
.32 and .38 Regulation Police revolvers
normally have "PAT. JUNE 5, 1917" stamped on the bottom of the extension stocks.