Unfortunately, the price to make any changes is exorbitant despite it being a 'flex' ticket.

I'm curious as to why there would be just a 2% chance of this working? Don't airlines have to rebook you (or they usually do it) if you miss a connection due to say, customs or baggage delay? What if I told them there wasn't enough time at JFK for a 3 hour connection (something that has happened before with me...) and I had to buy my own flight to LAS? Might they agree to not cancel the rest of the segments?