The definition that you use for the Lie derivative, and the result you wish to deduce, both hold for any contravariant tensor field, so I will address the question for this more general situation.
On page $321$ of Lee's Introduction to Smooth Manifolds (second edition), he defines the Lie derivative of a covariant tensor as you have done. On the very next page he has Proposition $12.32\ (d)$ which states (I'm paraphrasing):
Let $A$ be a smooth contravariant $k$-tensor field on a smooth manifold $M$, and let $V, X_1, \dots, X_k$ be smooth vector fields on $M$. Then
$\mathcal{L}_V(A(X_1, \dots, X_k)) = (\mathcal{L}_VA)(X_1, \dots, X_k) + \sum_{i=1}^kA(X_1, \dots, X_{i-1}, \mathcal{L}_VX_i, X_{i+1}, \dots, X_k).$
In Corollary $12.33$, Lee points out that this formula can be rewritten as (again, I'm paraphrasing)
$(\mathcal{L}_VA)(X_1, \dots, X_k) = V(A(X_1, \dots, X_k)) - \sum_{i=1}^kA(X_1, \dots, X_{i-1}, [V, X_i], X_{i+1}, \dots, X_k).$
This follows immediately once you know $\mathcal{L}_Vf = Vf$ for any smooth function $f$ on $M$ (this is Proposition $12.32\ (a)$), and $\mathcal{L}_VX = [V, X]$ for any vector field $X$ on $M$ (this is Theorem $9.38$).