The only thing you have to start over for is the "Night Terrors" achievement. This requires you to beat the original campaign on Ultra Nightmare difficulty. Unfortunately, there is no NewGame+ feature so you have to start from the beginning with none of what you gained from your original playthrough. This is also assuming you had already beat the game.
I would suggest just starting a new game on Ultra Nightmare and doing the DLC as you play through the game again. You get a new gun after the first DLC mission called the Nailgun and it makes life pretty easy. There are 3 types of ammunition for this gun. Nails, Rebars, and Railgun Slugs.
Nail ammo is fully automatic and...you guessed it...shoots nails at high velocity.
The Rebars shoot one at a time and are one hit kills to the common enemies. Headshots make crude wall decorations.
The Railgun Slug ammunition type comes with a scope that can see through walls. Line your target(s) up and shoot. Even through the thickest walls your target(s) will parish in wonderfully grotesque fashion.
You could always just use your original save file, change it to Easy, beat the DLC missions, and then start a new game on UN and ignore the new missions. I just chose to do it in one attempt and it hasn't been that difficult. Get good at the Jackpot game and you make some serious cash betting $50.
I hope this helps.