The only "rigging" is the frame, I guess - it gives you a bit more of a "yellow zone" before you lose the balance, making it more performable as a skill (in terms of reliability, safety, and theatricality,) and makes it easy to "hide" on a set. Chloe and I are writing a show right now that takes place in a parlor room, and that was one of the big reasons we designed the rig the way it looks.
Not sure lead would do much, honestly - the weight of you on top of the bottle would trump any amount of weight in the bottle. Also, I'm not sure what kind of safety equipment you could use, other than the occasional low ceiling and a helping hand.
That video of Daniela is so nice!! (She's a hell of a bounce juggler - holy smokes!) Those are pretty wide bottles - gives it a nice look. There are a few folks doing this kind of work that I've seen, and just about everyone uses a different technique and a different bottle.
Phoebe Carlson is a student (??) at NICA and she's using what look to be magnum champagne bottles (...check out this trick, holy hell.) http://iconosquare.com/viewer.php#/detail/1056450126553061261_2029151828
Shell Stachowicz is a ballerina who does it on pointe (kinda like that famous photo of the German girl doing the trick in the 30's) - http://iconosquare.com/viewer.php#/detail/1047836429596134371_1331054769
Smurf is a sideshow performer who does it on super wide-lipped milk bottles - http://iconosquare.com/viewer.php#/detail/1053067898396511581_4959770
Leigh Rhodes does it on the flat of her feet, on magnums - http://iconosquare.com/viewer.php#/detail/1091177578310756431_1833226369
Krin Haglund does it on magnums, too - also on the flats of her feet.