I think you would be better off with a length of 8mm diam silver steel, long enough to allow 2 bearing support with a spacer between the 2 bearings to provide better support, allow enough material at the front for your centre to be machined on the lathe.
Harden and temper before assembling and fitting into the adaptor to fit in your boring head tool socket.

Emgee

I feel very old now! When I did my apprenticeship live centres were about as common as rocking horse droppings! We had dead centres and a tin of tallow. I think I would stick that point in my boring head, spot of grease and off we go, if it's only one rod. If it's a run of them, ok it might be worth the effort.

Paul.

.

… and next to useless there, unless the chuck holds it concentric

MichaelG.

Really, I do it all the time, just skim the taper each time and don't remove from the chuck until the job is done, quite common practice. Though not much point is spending out on one when any bar end will do.

There is also the point that a 'live centre' should be soft material so it can be skimmed and the 'dead centre' hard so it doesn't get deformed by the job rotating on it. Until you let it overheat when it gets relegated to the other end.