Cannot drop Red Eagle sword having previously completed the quest

Solution 1:

Honestly? Don't worry about it.

Quest items have no weight - yes, even the ones from expired, completed, or broken quests, if they're bugged and stuck in your inventory, they also aren't adding to your encumberance. Unless you're really OCD about the item cluttering up your list, there's no harm in this bug.

Solution 2:

Use a weapon plaque.

I've not done the quest myself, but similar has happened to me with a dagger found in a cave for a master Alteration question in the College. Since it was found on a character that is a warrior, I will never do that quest. Now it hangs on the wall in Breezehome.

As long as you can equip the Red Eagle sword, you can dump it on a weapon plaque by putting it in your right hand (or both if a 2-hander) and activating the plaque with the weapon drawn.

If you don't own a house, you should probably be able to find a plaque in some weapon shops, or the Jorrvaskr.

EDIT: Apparently, non-owned plaques can't be used. You'll need to use your house's plaques.