The main issue here is that a gallon carboy/vessel is too small for fermenting a 1 gallon batch.
Fermentation usually creates krausen (foam), and will spew out, wherever it can, sometimes in an explosive way, making a big mess.
It's therefore generally advised to leave at least 1/5th of your batch size as headspace in the fermenter, where the foam can expand into.
In your case, a gallon carboy should contain no more than 4/5th of a gallon of wort/beer. It puzzles me that the kit instructions don't mention this. Instead of en airlock, you can use a blow off tube into a jar with water or Starsan to prevent some of the mess when some blow-off occurs, but you'd still lose any beer that's being blown off.
With some fermentations/yeasts a larger headspace than 1/5th is recommended. Higher fermentation temps also tend to create more foam especially in the beginning, and also require a larger headspace.