Since this question has come alive again, let me point out that the Hecke operators cannot give a splitting of $h^1(E)$ into two pieces over $F$, since the Hecke correspondences on a modular curve are always defined over $\mathbf{Q}$.
(If you allow some extra stuff like the Atkin--Lehner operator on $X_1(N)$, then you can get some slightly larger fields showing up, but they will still be totally real, and thus not capable of "seeing" the splitting of $h^1(E)$.)
If you take a higher-weight CM cuspform $f$ of weight $> 2$, associated to some Groessencharacter $\Psi$, then the situation is even worse: you can define a motive associated to $f$ using Kuga--Sato varieties, and a motive associated to $\Psi$ using the product of $k-1$ copies of an elliptic curve with CM by $K$. These motives really should be the same, because their $\ell$-adic Galois representations are the same for every $\ell$, but there is (as far as I know) no natural way of writing down a correspondence that gives an isomorphism between them in the category of Chow motives.