Long Haul aircraft carry shed loads of champagne, it goes on and off in the drinks barcarts, and those take a heck of a beating.
The hold of an aircraft is pressurized, at the same pressure as the cabin.
I would be a bit concerned carting champagne around in my luggage, mainly because it gets messy if the bottle explodes. In your case I would use a separate hard suitcase, wrap the bottles with bubble wrap, tape it up securely, and then enclose them in a couple of heavy duty bin bags. Anything else that goes in that case should be easily washable.
If you put a couple of frozen freezer blocks in at the last minute, then they’ll probably arrive cool. Hmm a few large tupperware boxes would also be a good idea.
I would not be that worried about them being stolen in transit, nobody will see the labels, or know what is in the suitcase - sure it will be scanned behind the scenes, so you just might attract some interest from security.
Personally I would baulk at paying $600 for a bottle of champagne, but I’m a cheapskate and know how much airlines really pay for the stuff.
Incidentally, you could easily do a test run with a couple of bottles of NV, wrap them up and throw the whole lot around in the back garden, leave it in the sun, and drive around with it in your car boot for a day.
I once worked for a company that was sending out large numbers of delicate glasses, they got various polystyrene packing options and played football with them in the office. Another time they tested packaging for international shipping of jars of jam, by sending it round Germany on the rail mail system.
Incidentally, you can get polystyrene shaped boxes for wine, I’ve had wine sent to me in those, but I would stick with the bubblewrap as it is easier to get hold of.