I don't have any special insights into this, but my first thought was to just turn the lamp upside down and fill the base with plaster of paris. That should give it sufficient weight to be stable and it should stay in place reasonably well.
Only issue that comes to mind is that you may have to figure a way to make a channel for the electrical cord; of course you can always mold the cord into the plaster, but for future reparability I'd think you'd want to be able to remove the cord. Maybe put a sheet of saran wrap over the top then use a pencil to press a channel into the plaster before it sets? With the plastic wrap you should be able to remove the pencil once it's hardened.
I'll be interested to see what you come up with as a solution. Good luck with your repair!