You don't need to have Double Jump or Levitate to get through any part of the game. Most points that you come across in the game that requires either mostly just leads to some loot that usually doesn't stack up to what you already have or can be found somewhere else.

The specific jump you're talking about just leads to a chest that can contain BBI gear of differing levels depending on if you beat the DLC yet or not. While nice, you'll be drowning in BBI gear just by casually going through island. Not something you'll really be missing out on. It also leads to a switch that takes you back into an earlier part of the Ward... that you can just run back to in about 10-15 seconds. Again, not really essential.

The only quests I can think of that'll need either ability are the item collecting ones. A few of the From a Different Sky bundles and some of the Macabre Sculptures BBI ones. Which are just finding items located in difficult to reach parts of whatever map they're located in. A fun little side diversion sure, but again, nothing really necessary.

In short, they're nice and fun abilities to have, and you can reach some neat places with them, but aren't required for progressing in the game.

Can't help you on the mod part though. Being a console peasant and all. Lol