With permission, here is a response from Rick Kerschner, previously of the Shelburne Museum:
“We have run into similar problems at Shelburne. I agree about not wetting the wood of a collection piece. Think about supporting the barrel slats from the inside. Cut a circular piece of 2” or 4” thick Ethafoam to be slightly larger than the circular base of the inside of the churn, perhaps 1/2” larger in diameter. Cut the disk in half so you can get the disk into the churn. Press the two halfs together at the base of the churn to push the stays outward and into place. You may have to trim the disk several times to get it to stay in place. Too big initially is better than too small. You could also try the 4” thick grey stiff polyester foam used for packing paintings. You may not have to cut that in half to get it into the churn because you could slightly “fold” it. You could do the same in the middle (the widest point?) and at the top of the churn, but it will be more difficult because the foam will tend to fall toward the bottom. If truly a barrel shape, a disk will stay in place at the widest part of the middle. A churn shape that is simply a taper from narrow to wider circle will be more difficult to support from the inside. You might have to hold the top disk in place with a few screws or nails through the cracks between the stays into the foam and resting on the top of the outer steel rim.
Depending on just how shrunken the stays are, you may need 6 or more hands to hold them in place while fitting the foam inside.”
I hope that helps!