I'm trying to create an enchantment for a sword that does 20 points of fire damage per strike regardless of what level my character is. Is there any way to do this exclusively for custom items and enchantments, without having to tamper with global settings of the game?

Create just one enchantment and one weapon and that should do the trick. Enchantments placed in the CK are "hardcoded" in a way: meaning if in the CK you put an ecnhantment on a weapon doing 20 points of fire damage that weapon will allways do 20 fire damage per strike no matter what the player's level or enchanting skill is.

The leveled weapons you find in the game all use the so called leveled item lists. To create those different versions of the weapon each with it's own enchantment must be created and added to the list. It is not done by some formula that calculates damage based on the level. Even the base damage of a weapon allways stays the same.

Resulting damage can be affected by Spells/perks/abilities and combat skill level. The effect of the skill being the only thing that isn't changeable using the CK as far as I know.