Look for makers or other id marks on the brass ends of the web sling.  If it says 'Orndorff' with other details, then keep it on the Krag.  It could be an original 'Tropical' sling which would be correct for your rifle. 

Oddly enough, the army was not terribly particular about what sling (treated simply as an expedient carrying tool) was used on what rifle.

Krags are thus "correct" with any current, or previous, sling that was in serviceable condition. There is some lack of agreement as to whether the Mills web item was actually originally intended for use as a gunsling, or for some other piece of equipment, but it looks good, and fits, so a number of people - including myself - display specimens on rifles.

The Kerr sling is a WW1 period item, hence not around in the heyday of the Krag - but - the use of Krags in that conflict (once derided as false) has now been legitimatized. While a combat role was never the intent, it can no longer be denied that some Krags DID go overseas - so, even the Kerr sling is not out of the picture at all.

Most school guns have a block JFC cartouche, without date, but apparently some, made up at facilities other than Springfield, retained the original stock markings. Also, more seem to be found with the 1901C sight, than the 1902 shown.

Thanks for clearing that up. Only the first of many revelations to come, I hope!