There is no hope. Timothy Budd already explained why staying on the sphere $S_t$ of possible locations of $M$ will not work; so this is to explain why leaving it will not help. What is enough to show is the following claim: at large times, if $t_1<t_2$ and $C_{t_{1,2}}$ are at $\epsilon$-neighborhoods of $S_{t_{1,2}}$ respectively, then the radial projections of $C_{t_{1,2}}$ onto the unit sphere are at distance at most $c\cdot \log {t_2/t_1}$ from each other. With the claim, essentially the same argument will work.

But the claim is fairly obvious: between $t_1$ and $t_2$, $C_t$ cannot get more  than $(v_c-v_m)(t_2-t_1)+2\epsilon$ behind $S_t$, otherwise it wouldn't be able to catch up. If $t_2-t_1<t_1\cdot \frac{v_c-v_m}{100v_m}$, then this means that $C_t$ is always at distance at least $\frac12v_m t$ from the origin, and then the speed of the projection is bounded by $\mathrm{const}/t$ as in Budd's answer. If $t_2/t_1\geq 1+\frac{v_c-v_m}{100v_m}$, then we can just bound the distance between projections of $C_{t_{1,2}}$ by $2$, and take the constant $c$ to be $2\cdot \log\left(1+\frac{v_c-v_m}{100v_m}\right)$