Is it the whole chamber or just the throat? If stuffing bullets deeper into cases allows feeding, it sounds like the cases fit into it fine, which would mean the chamber is deep enough, but the bullets are stopping prematurely in the throat. I suppose it is conceivable you got a chamber that was roughed but did not get the final finishing reamer applied. Roughing reamers typically have no throat portion. It is also possible the manufacturer just had a reamer design that has a short throat or was using a normal reamer that had been resharpened one time too many.
If you drop a finished round with normal seating depth in rather than feeding it, does the head of the case fall in flush with the back of the barrel? Does that happen with commercial ammo? If so, then its not necessarily the chamber or the throat that's out of spec, but some other aspect of feeding that's the issue. The chamber being too narrow from using a worn reamer or improperly contoured chamber mouth or any of the various timing issues that cause feed problems could be involved. That would need a gunsmith's skills to check.
You ought to be able to send the barrel to S&W for inspection. You could also have a gunsmith apply a match throating reamer to it. Indeed, if you have a local gunsmith with a standard 9 mm reamer, just touching the chamber with that may fix the throat without deepening the case portion.
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