This is an untested idea, as far as I know, but it ought to work.
1.) Mount a short piece of your 1 1/4" stock on the lathe (between centers
or in a chuck.)
2.) Attach wooden pieces to it, with a hose clamp, and turn the outside of
the attached pieces round.
3.) Mount your 1 1/4" spindle blanks between centers, and attach the rounded
pieces in the middle with a hose clamp.
4.) Fit the steady rest over the round blocks.
5.) Turn the spindle to completion on both sides of the steady rest.*
6.) Move the steady rest to a finished area near the middle of the spindle,
and finish up.
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*You could remove the blocks/hose clamp and move the steady rest as soon as
you have a round area near the middle. Of course, leave any 1/2" parts to
be done last.