The NAS (or whatever its current title is, sorry!) has records for Scottish railways. Even in England & Wales, while much went to Kew, other staff records can be found in county record offices.
Dundee seemed to be served by the North British Railway Co and the Caledonian Railway Co (plus a joint operation that I'm guessing was jointly worked by those two). According to David Hawkings' "Railway Ancestors" the situation doesn't look optimistic for staff related records for the "Caley" in that era, but the NB looks a bit more interesting. In particular there's BR/NBR/4 Pieces 2 to 11 at Edinburgh, described as "Accident Books 1869-1915". These just might cover all staff accidents, rather than statutory reports for train accidents. Worth a punt. There are other staff books for the NB, though they're described as "Central Section" or "Southern Section", etc and I've no idea what the NB's sections were....
(Check the NAS Catalogue - BR/NBR/4/2 at least is described as being held off-site.)
As you suggest, Dundee Archives may know a bit more.
