Is it not simply a conversion of an iterated homogeneous equation into a non-iterated non-homogeneous equation? By hypothesis, $L_m u = v$, where $L_k v = 0$. Every such $v$ can be obtained from some initial data at $t=0$. On the other hand, every every such $u$ can be obtained by solving the inhomogeneous linear equation, using advanced and retarded Green functions for $L_m$, plus some solution $w$ of the homogeneous equation $L_m w = 0$. Then the freedom in the choice of $w$ is fixed by matching the initial conditions for $u$ at $t=0$. Thus, to solve the iterated problem, you need only know how to solve the homogeneous and inhomogeneous non-iterated problems.

Are you perhaps looking for some explicit formulas for the Green functions that match initial conditions to the solutions that result from this procedure?

**Update.** About the choice of initial conditions. BTW, I didn't immediately pay attention to the fact that you want the initial data specified at $t=0$, which is the location of the singularity of the coefficients of your PDE. This is a kind of Fuchsian equation that requires extra care when imposing boundary conditions. I haven't examined this example in detail yet, but you may not be able to specify all the derivatives, $0\le i \le 3$, while requiring the solution to stay bounded at $t=0$. First, let me illustrate how to determine all the initial conditions when $t=0$ is a regular value for the coefficients. I may come back to the singular case later.

Write your solution as $u = u_v + w$, where $L_m w = 0$, $L_m u_v = v$ and $L_k v = 0$. You are free to set the initial conditions $D_t^{0,1}w(0,x) = u_{0,1}(0,x)$ and $D_t^{0,1} u_v(0,x) = 0$. Then the first two initial conditions are met, $D_t^{0,1}u(0,x) = u_{0,1}(0,x)$. On the other hand, just by taking time derivatives you find that $D_t^{0,1}v(0,x) = D_t^{0,1}L_m u_v(0,x) = D_t^{0,1}L_m u(0,x)$, where the right hand side can be determined from the knowledge of the derivatives $D_t^i u(0,x)$, $0\le i \le 3$. This gives you the initial conditions for solving $L_k v = 0$.