The map in question is $f: e^{\theta^IT_I}\rightarrow e^{M^I_J\theta^JT_I}$, for $M$ an invertible $8\times 8$ matrix and $T_I$ the half Gell-Mann matrices which generate $SU(3)$. My reasoning is as follows:
The map $\theta^I \rightarrow M^I_J\theta^J$ is smooth and invertible with smooth inverse.
The map $\theta^I \rightarrow e^{\theta^IT_I}$ is smooth and invertible with smooth inverse.
Hence, the map $f$ is a diffeomorphism of $SU(3)$.
Is this reasoning correct?