The maximum likelihood estimator (MLE) for $(\mu_1,\mu_2)$ is $(X,Y)$. So, by the functional invariance of the MLE (that is, simply by definition), the MLE of $g(\mu_1,\mu_2):=(\mu_1+\mu_2)/2$ is $g(X,Y):=(X+Y)/2$.

The maximum likelihood estimator (MLE) for $(\mu_1,\mu_2)$ is $(X,Y)$. So, by the functional invariance of the MLE (that is, simply by definition), the MLE of $g(\mu_1,\mu_2):=(\mu_1+\mu_2)/2$ is $g(X,Y):=(X+Y)/2$, which also, obviously, maximizes the profile likelihood $$L_{X,Y}(\mu):=\sup\{L_{X,Y}(\mu_1,\mu_2)\colon(\mu_1+\mu_2)/2=\mu\}$$ in $\mu$, where $L_{X,Y}(\mu_1,\mu_2)$ is the likelihood.

One may note that the MLE $(X+Y)/2$ of $(\mu_1+\mu_2)/2$ does not depend on $(\sigma_1,\sigma_2)$.