$M$ is smooth, so there is a cellular decomposition with a single 4-cell. Thus we want $TM$ trivializable over the 3-skeleton. We analyze sections of the frame bundle, with group $SO(4)$. Obstruction theory says we can extend over the 2-skeleton, because the class is $w_2\in H^2(M; \pi_1SO(4)=\mathbb{Z}_2)$ and you assumed that to be zero. The next obstruction (extending over the desired 3-skeleton) is automatic, since $\pi_2SO(4)=0$.