I'd reccomend keeping Bite on your Eevee for the long run, due to the coverage for Sabrina's Gym and the tower in Lavender Town.
Like you said, Headbutt is good STAB and Quick Attack has lots of PP plus priority.
So this leaves Double Kick. Early on, it's extremely useful, but as you progress it gets less useful, as normal types don't appear as often. I'd reccomend replacing Double Kick with one of the partner move tutors's moves. If I had to reccomend one move for you, I'd reccomend Bouncy Bubble, as it's a solid water type move that also heals you. There's lots of rock/ground type trainers in the game, and caves coming up, so it'll also help with some type coverage there.
I hope this helped!