Hi Jon
My tailstock self ejects so I normally give the centre or centre drill a good shove when putting it in, it is 3 Morse taper. I keep it and all tapers that go into it scrupulously clean and almost dry so I get a good grip, I don't oil them, and just before I insert them I clean the hole with my finger and drag the taper on the tool through a clenched hand to check for any dirt.
The tailstock barrel is keyed so it cannot rotate and it has a lock to make it rigid, the turning forces should keep it against the key if it is loose in the key anyway and there is no lock. If your tailstock has a through hole you could use a draw bar.
I would not recommend this method for heavy or high accuracy work, the rigidity is not as good as using the tailstock. for small pieces it is fine.
I should have mentioned the boring bar movement axis should be horizontal, or you will not get a straight taper.
Regards
John