How to compute this exponential matrix
The idea is indeed to compute $M^{-1}e^{[S]\theta}M$. The reasoning is as follows. Recall that the exponential map of matrices is defined by its power series: $$e^A=\sum_{n=1}^\infty \frac{1}{n!} A^n. $$
Note that for all $n$, $$(M^{-1}AM)^n = M^{-1}AM \cdot M^{-1}AM \cdot ... \cdot M^{-1}AM = M^{-1} A^n, M$$ which implies $$e^{M^{-1}AM} = M^{-1} e^A M. $$ Apply this with $A = \theta [S]$ to see that you need to compute $M^{-1}e^{[S]\theta}M$ to find $e^{M^{-1} [S]\theta M}$.