Patch definitely worked in the way that now the deck doesn't get ruined when opened via Web Decker.
But exporting still didn't work. Renaming all of the cards also didn't help,
Eventually, I figured that the problem was in the first card's script. I wanted it to play a sound file on a loop during the title screen, and this made HTML file unplayable for some reason. Deleting the script solved the issue. I attached the sound to the "begin" button, so it's a decent compromise. :)
Thanks a lot for the assistance!