Springfield's production reports are tabulated on page 256 in the book. From their annual production reports. They list exactly 1 made in FY02-03.
Strange thing is I encountered that 1. How's that for needle in a haystack?
With that data point, is it possible that more than 1 received later cartouches? Certainly. If the, say, Navy wanted 200 of them and wanted them inspected SA would do so. So clearly within the realm of possible and within the realm of what's been observed from them.
The bulk of them were in the three blocks. Outside of those it's the typical odd dribs and drabs for reasons only they were clear on.
Is it possible that they'd put 1902 sights on small dribs and drabs? Well, they did make those sights for something right?
The missing piece is an unexpected personnel change. Two in fact. M-1899 carbines past the 1901 production are oddities. 1 new manufacture per their record. Anything else is a redo of an existing gun.
So why no carbines of note in 1902 or 1903? "The regulars were going to get the M-1902 (which became the M-1903)." The Militia? "Refused to maintain mounted troops in significant numbers due to the cost of maintaining horses." Infantry is cheap. Mounted not so much.