Is it legal/possible to request a flight-path change from Domestic to International minutes after take-off, for a privately owned jet?
Generally speaking, ATC doesn't care who you are or where you're going. Their only job is "a safe, orderly and expeditious flow of traffic." If you request to amend the destination on your flight plan, they'll just type it into their computer and give back the new clearance it spits out. Diverting is a fairly routine thing, actually, especially when bad weather is afoot. If they even ask why, it's to make sure it's not due to an emergency or urgency. What (if anything) happens once you're safely on the ground at that new destination is someone else's problem.

That said, the militaries of many countries may take exception to someone flying into their airspace without prior permission, which is a much more complicated topic.