Listen to Sunny. The contact itself can easily spark off a complete destruction of the tribe, no matter how well intentioned you are. I'm not saying put up razorwire around their land and shoot anyone who tries to contact them, but any contact needs to be extremely controlled and cautious. If you simply load these people on a plane and fly them to Chicago or London and start showing them around, you are going to destroy them. I don't know what in history makes you think the result would be different from the last thousand times modern people have done this.I'm not sure they'd understand that choice. I doubt whoever makes contact will bring along a toilet and a brush to demonstrate. I'm guessing the pitch would be all the wonders of the world followed by a harsh reality of crap and firewater. This is a legal issue that is struggled with a lot in the US in regards to medical treatment. Even when you present all the information people have a tendency to just hear "you'll get better and be fine" especially when they don't have a solid grasp of medical terminology and the like.
Of course society somehow could offer them an aid package during entry. But making society pay for others is a whole different issue you can discuss with Teaos.
Also, likely, you're only giving half of them a choice. The others who wanted to stay will be left in a society that just lost half its membership, and may no longer have the genetic diversity or diversity of skills to survive. Particularily if its the younger members leaving, as is often the case.
Also there isn't any way, really, to shut off the disease issue. Though there are things that can be done to help.
If you need relatively "modern" examples, then go look up the Hmong people on Wikipedia. And they weren't anywhere near as isolated as this newly discovered tribe, but the results were still basically disastrous.