As requested in the comment, let me explain what I could extract from the literature on this problem. (This is not the solution asked for in the OP, but what I have would be too long for a comment.)

The potential in the OP is known as the "Razavy potential" or "double cosine potential". A recent study is Exact Solutions of the Razavy Cosine Type Potential (2018). The Schrödinger equation is $$-\psi''(x)+V(x)\psi(x)=E\psi(x),\;\;V(x)=\tfrac{1}{4}\xi^2\sin^2 x-(a+1)\xi\cos x,$$ on $-\pi\leq x\leq\pi$ with $\psi(\pm\pi)=0$. The differential equation in the OP is for $\xi=2h$ and $a=-3/2$. (I note that the cited paper assumes $\xi,a>0$, but that does not seem to be an essential condition on the solution.)

The substitution $\psi(x)=\exp(\tfrac{1}{2}\xi\cos x)\phi(x)$ and the change of variables $z=\cos^2(x/2)$ produces a confluent Heun differential equation with solution given by the Heun function $$\phi(z)=H(2\xi,-1/2,-1/2,-(2a+1)\xi,2(a+1)\xi+3/8-E;z).$$ The energy $E$ should then be obtained from the boundary condition, but the cited paper does not succeed in obtaining a closed-form solution.

As requested in the comment, let me explain what I could extract from the literature on this problem. (This is not the solution asked for in the OP, but what I have would be too long for a comment.)

The potential in the OP is known as the "Razavy potential" or "double cosine potential". A recent study is Exact Solutions of the Razavy Cosine Type Potential (2018). The Schrödinger equation is $$-\psi''(x)+V(x)\psi(x)=E\psi(x),\;\;V(x)=\tfrac{1}{4}\xi^2\sin^2 x-(a+1)\xi\cos x,$$ on $-\pi\leq x\leq\pi$ with $\psi(\pm\pi)=0$. The differential equation in the OP is for $\xi=2h$ and $a=-3/2$. (I note that the cited paper assumes $\xi,a>0$, but that does not seem to be an essential condition on the solution.)

The substitution $\psi(x)=\exp(\tfrac{1}{2}\xi\cos x)\phi(x)$ and the change of variables $z=\cos^2(x/2)$ produces a confluent Heun differential equation with solution given by the Heun function: $$\psi(x)=\exp(\tfrac{1}{2}\xi\cos x)H\bigl(2\xi,-1/2,-1/2,-(2a+1)\xi,2(a+1)\xi+3/8-E;\cos^2(x/2)\bigr).$$ The energy $E$ should then be obtained from the boundary condition $x=\pm\pi$, so at the origin for the Heun function, but the cited paper does not succeed in obtaining a closed-form solution.