I'd go small laptop (carry-on) with very-large LCD (check-in, with original packing material). But that's just me.

If you don't have the original packaging, pay the $50 for a pack-and-ship location to do a custom spray-foam box (not all of them have the equipment, you have to call around). Done properly, spray-foam could practically be dropped from a moving plane, and what's inside would survive.

If neither of those options are doable, expect a broken monitor when you get to your destination.

Steve B.