The clamp in the sketch seems a good idea but I would favour supporting the other end too to ensure it does not go out of line. I would not hold anything by hand when using a Forstner bit or similar. If the bit grabs it can throw the work out for your hands and maybe into your face! I was lucky to dodge the wood and not get my fingers drawn into the bit on the only occasion I tried this! Never again.
If the parts are costly then it would be worth using such a clamp fitted on the faceplate of a lathes to bore the centre away. Find a friendly model engineer with a lathe!
To find the centre of a circle for practical purposes these gadgets are the thing.....
The first is a simple plastic give away device used by woodturners and the other a Moore & Wright No 825A for engineers. Both use the same principles, 2 lines are drawn at about right angles and a third to check for errors, they should all coincide at the centre. There is also 'jenny leg' (odd leg ) dividers for the same purpose where the non pointed leg goes on the outer surface of the cylinder to scribe the arcs.
[img][/img]