Jerry,
You won't have much of the bushing in the collet, so you might encounter runout problems. Also, you'll be using your chisels very close to the headstock, which may be inconvenient. Lee has already mentioned that the live center may not align well with the other bushing.
I've modfied some of my bushings for turning between centers. I chuck up each bushing backwards in a collet chuck, and using a 60 degree center bit, chamfer the hole to match the 60 degree centers. Now they mate truer. To get really picky, chuck stepped bushings on the portion that slips inside the tube, not the outside. That way, if the two circumferences are not concentric, you'll be drilling the chamfer relative to the more important one.
This improves stock bushings, but the custom made ones are still nicer.
I hope that helps,
Eric