Very nice design and workmanship!
In the pursuit of a perfect jaws/fixture design to hold a barrel and allow it to pivot freely as it is adjusted at the other end by the spider I think the perfect solution would be some type of a ball and socket design where the ball had a hole large enough for the barrel. Some method to lock the barrel to the ball would be needed and this fixture would then go in between the four jaws of a chuck. Some way to keep the ball from spinning freely with respect to the chuck, would also be needed.
I think that all fixtures with jaws that aren't of this ball and socket-type must allow for some small amount of sliding motion of the barrel to occur while clamped between the chuck jaws and while the spider is adjusted. Therefore, the jaws cannot be so tight so as to prevent this small amount of sliding without possibly some small bending of the barrel.
Admittedly this ball-and-socket fixture design, while maybe close to a perfect solution, is likely more complicated than what is necessary for the work needed in centering a barrel, etc., and achieving good results.