Charlie,
I don't have a small chuck to trade to you but if I understand correctly what you need to be able to do, I'd like to suggest an alternative solution. The problem with holding the ID of a bushing with a 4-jaw chuck so that you can machine the OD is that if the bushing doesn't have a pretty heavy wall thickness, tightening the chuck inside of it could distort the bushing at the four points where the jaws are touching. Then you turn the OD down a little bit and everything will look fine until you loosen the 4-jaw enough to rotate the bushing 45 degrees, and re-tighten the chuck. If you then indicate the OD of the bushing you will find that it is not perfectly round.
To avoid that happening, you can use an ID expanding mandrel. At any given time, you will usually find a relatively cheap set of expanding mandrels for sale on eBay. They are of course made in China but if you hold the 1-1/4" diameter one in your 3-jaw chuck, mark where Jaw #1 is in case you need to remove it from the chuck and later reinstall it, and turn the mandrel down with the adjustment screw just snug until you can just slip the bushing over it, and then tighten the mandrel, you will get a part that is not slightly egged and an indicator on the OD should show zero run-out.