Thank you so much guys for your input! I couldn't believe I already had 19 replies within 9 hours of posting! That's incredible. How many forum members are here?
I've carefully read each of your answers. I guess I will be well putting the whole guitar in the suitcase with clothes.
Luckily, I am not travelling with United (United breaks guitars LOL!). For the 335 I bought a CasExtreme. Those cases started being manufactured as a response to that United breaks guitars video. So I hope it works.