Granpa's Gun Parts is the best place to get what you need from an honest supplier. His father sold Krag and Springfield parts before him and he, somehow, still has parts to sell. If he says it's original, it is. I have bought from him before. There are some reproduction oilers and cleaning rods out there as you suspected.

The oiler is usually unmarked. The butt place holds the oiler, which is slid in first into the slight cut out recess towards the curved side of the butt, then the three part cleaning rod pieces fit neatly into the three holes deep in the butt which you can see if you shine a flashlight into the butt compartment.

So, you have it correct. The earliest Krag, M1892, did not have a butt compartment and held a full length ram rod (cleaning rod) under the barrel. It had no oiler that fit into the rifle anywhere.

Now you need the cleaning rod set. The full set of cleaning rods and oiler usually cost around $50. (There are two extension segments to the three cleaning rods and one upper segment that has the patch slot. make sure you are getting the correct pieces if you buy them.)

The correct oiler may have a point on the stem connected to the unscrewed cap - not really a "brush".

A helpful tidbit:  Once you get your cleaning rods, and load up your butt trap, stuff some cleaning patches in there before you close the trap.  This will keep everything nice and tight and prevent things from rattling around!

When assembled, the three rod sections are too short to clean the 30" Krag rifle barrel.

In the old days, soldiers would 'pair up' and combine four sections, in order to clean their Krags in the field.

Reportedly, carbines only carried two rod sections, that were slightly heavier than the rifle version.