Isolating a variable trapped in trigonometrical function $\frac{p}{r}=\pm1 + \varepsilon \cdot\cos\sqrt{K\frac{M}{r^3}t^2}$. Is there any technique or simple way to isolate $r$ ?

There's no way to do that. One thought some people have is to use complex arithmetic and try the Lambert W function, but as has been shown in other posts here, allowing complex numbers throws the equation off. You will have to solve numerically.