+1 to what Jason says about capstan and repetition lathe tooling.
That problem was solved many, many years ago to make "end-on" manufacture of accurate components possible. Since then the number of components produced on capstan and repetition lathes is way beyond convenient, or even inconvenient, counting.
Time to start digging in the historical records to find something simple to make, easy to set up and effective. Modern box tooling tends to be a bit overkill and somewhat specialised for small ranges of sizes.
Easiest would be a simple disk with a slot in it to carry a short length of HSS as the tool bit. Slot needs to be angled at the back so the side of the tool bit cannot come into contact with the job and mar the face. If working brass you can probably get away with no rake as the cut is shallow. Tool setting becomes much easier.
As the cutting tip has to be on centre any top rake makes life more complicated. Back in the day tool setting was a job for a specialist.
I'd just set up stops and have at it "side on" in the normal manner with an old fashioned wide parting tool. Of which I have several in stock which mostly get used for this sort of thing. But my lathes are much more industrial than yours.
The modern professional uses a diamond shape insert.
Clive