The answer is 'yes, we can'.

Since we are in the simply-connected case, by the deRham Theorem, we can assume that $(M_i,g_i)$ for $i=1,2$ are isometric to $$ (\mathbb{C}^m,h_0)\times (N_1,h_1)\times\cdots \times (N_k,h_k) $$ where $h_0$ is the standard flat metric on $\mathbb{R}^{2m}$ and, for $1\le \ell\le k$, $(N_\ell,h_\ell)$ is an irreducible symmetric space that has a $h_\ell$-parallel, $h_\ell$-orthogonal complex structure $A_\ell$. (I can't use $J_\ell$ because $J_1$ and $J_2$ have already been taken. In fact, because the holonomy group of $h_\ell$ acts irreducibly on the tangent space at any point with commuting ring isomorphic to $\mathbb{C}$, the only $h_\ell$-parallel, $h_\ell$-orthogonal complex structures on $N_\ell$ are $A_\ell$ and $-A_\ell$. It follows that we can assume that $$ (M_1,g_1,J_1) = (\mathbb{R}^{2m},h_0,A_0)\times (N_1,h_1,A_1)\times\cdots \times (N_k,h_k,A_k), $$ as Hermitian symmetric spaces while $$ (M_2,g_2,J_2) = (\mathbb{R}^{2m},h_0,B_0)\times (N_1,h_1,\epsilon_1A_1)\times\cdots \times (N_k,h_k,\epsilon_kA_k), $$ for some choice of $\epsilon_i = \pm 1$ while $A_0$ and $B_0$ are orthgonal complex structures on $\mathbb{R}^{2m}$ (not necessarily inducing the same orientation on $\mathbb{R}^{2m}$, of course).

The two flat factors are clearly isometric as Hermitian symmetric spaces, so the only question is whether there is an isometry $c_\ell:(N_\ell,h_\ell)\to (N_\ell,h_\ell)$ that satisfies $c_\ell^*A_\ell = -A_\ell$.

This may not be immediately obvious from the definitions, but it can be proved abstractly from the standard symmetric representation as $G/U$ or simply checked case by case using Cartan's classification of the irreducible Hermitian symmetric spaces.

For the cases AIII (the complex Grassmannians and their duals, including the projective spaces) and BDI (the complex quadrics and their duals) there is an obvious anti-holomorphic isometry.

For the case DIII, the set of positively oriented orthogonal complex structures on $\mathbb{R}^{2n}$ and its noncompact dual, an antiholomorphic involution (in the compact case) is conjugation with an orientation-reversing isometry.

For the case CI, the set of complex Lagrangian subspaces of $\mathbb{C}^{2n} = \mathbb{H}^n$ (and its dual), an antiholomorphic involution (in the compact case) is simply taking the orthogonal complex Lagrangian subspace.

In the first exceptional case EIII, one can rely on the fact that $\mathrm{E}_6\subset\mathrm{SU}(27)$ is defined as the stabilizer of a cubic form $C$ with real coefficients (as per Cartan), so complex conjugation preserves the singular locus of the cone $C=0$, and EIII is just the projectivization of this singular locus, a complex manifold of dimension $16$ that is invariant under the conjugation.

The second exceptional case, EVII (compact type), a complex manifold of dimension $27$ has is a similar description as a projectivized orbit of a singular locus of the quartic form Q with real coefficients on $\mathbb{C}^{56}$ stabilized by $\mathrm{E}_7\subset\mathrm{Sp}(28)\subset\mathrm{SU}(56)$. See Cartan's paper on the classification of the real forms for details.

The answer is 'yes, we can'.

Since we are in the simply-connected case, by the deRham Theorem, we can assume that $(M_i,g_i)$ for $i=1,2$ are isometric to $$ (\mathbb{C}^m,h_0)\times (N_1,h_1)\times\cdots \times (N_k,h_k) $$ where $h_0$ is the standard flat metric on $\mathbb{R}^{2m}$ and, for $1\le \ell\le k$, $(N_\ell,h_\ell)$ is an irreducible symmetric space that has a $h_\ell$-parallel, $h_\ell$-orthogonal complex structure $A_\ell$. (I can't use $J_\ell$ because $J_1$ and $J_2$ have already been taken. In fact, because the holonomy group of $h_\ell$ acts irreducibly on the tangent space at any point with commuting ring isomorphic to $\mathbb{C}$, the only $h_\ell$-parallel, $h_\ell$-orthogonal complex structures on $N_\ell$ are $A_\ell$ and $-A_\ell$. It follows that we can assume that $$ (M_1,g_1,J_1) = (\mathbb{R}^{2m},h_0,A_0)\times (N_1,h_1,A_1)\times\cdots \times (N_k,h_k,A_k), $$ as Hermitian symmetric spaces while $$ (M_2,g_2,J_2) = (\mathbb{R}^{2m},h_0,B_0)\times (N_1,h_1,\epsilon_1A_1)\times\cdots \times (N_k,h_k,\epsilon_kA_k), $$ for some choice of $\epsilon_i = \pm 1$ while $A_0$ and $B_0$ are orthgonal complex structures on $\mathbb{R}^{2m}$ (not necessarily inducing the same orientation on $\mathbb{R}^{2m}$, of course).

The two flat factors are clearly isometric as Hermitian symmetric spaces, so the only question is whether there is an isometry $c_\ell:(N_\ell,h_\ell)\to (N_\ell,h_\ell)$ that satisfies $c_\ell^*A_\ell = -A_\ell$.

This may not be immediately obvious from the definitions, but it can be proved abstractly from the standard symmetric representation as $G/U$ or simply checked case by case using Cartan's classification of the irreducible Hermitian symmetric spaces.

For the cases AIII (the complex Grassmannians and their duals, including the projective spaces) and BDI (the complex quadrics and their duals) there is an obvious anti-holomorphic isometry.

For the case DIII, the set of positively oriented orthogonal complex structures on $\mathbb{R}^{2n}$ and its noncompact dual, an antiholomorphic involution (in the compact case) is conjugation with an orientation-reversing isometry.

For the case CI, the set of complex Lagrangian subspaces of $\mathbb{C}^{2n} = \mathbb{H}^n$ (and its dual), an antiholomorphic involution (in the compact case) is simply taking the orthogonal complex Lagrangian subspace.

In the first exceptional case EIII, one can rely on the fact that $\mathrm{E}_6\subset\mathrm{SU}(27)$ is defined as the stabilizer of a cubic form $C$ with real coefficients (as per Cartan), so complex conjugation preserves the singular locus of the cone $C=0$, and EIII is just the projectivization of this singular locus (a complex manifold of dimension $16$), which is invariant under the conjugation.

The second exceptional case, EVII (compact type), a complex manifold of dimension $27$, has a similar description as a projectivized orbit of a singular locus of the quartic form Q with real coefficients on $\mathbb{C}^{56}$ stabilized by $\mathrm{E}_7\subset\mathrm{Sp}(28)\subset\mathrm{SU}(56)$. See Cartan's paper on the classification of the real forms for details.