A quick trip to the party store this weekend provided me with a variety of balloon styles to play with. The best candidate so far is a "punch-ball" balloon, which is both much thicker than a normal balloon and is also pleated. The fill tube extends into the neck of the balloon right down to the tip, which makes it easier to insert the whole assembly. Also, before inserting the balloon, I pop in a short length of polyethylene tubing that serves as a bypass and allows soda trapped under the balloon to escape.
The system permits quite precise deployment of the soda, although I am still considering implementing a flow restrictor. In the second image, I've deployed soda most of the way up the tube but halted it at that point. By fully expanding the balloon I can expel almost all the soda, although in practice I'll only need to display a few fluid ounces.
One possible improvement might be to use a smaller bottle and a tube sized so that when the bottle is completely empty, I have a soda globe of exactly the right size.