Congrats on your new baby and welcome to the forum.
I've some experience traveling with instruments - mostly violins which are a little smaller of course, but also guitars. It depends on the airline and if the flight is full or not, but most will allow you to take your baby as hand luggage - one of the flight crew can put it in their locker during the flight. Getting past check-in is the crucial part - just explain how valuable and fragile your guitar is and how you're usually allowed to take it on board with you. Usual rules apply - just be polite and flutter your eyelids a bit all the way from check-in to boarding
