In a two block pulley system, do we consider tension to be an internal or an external force, in order to find the acceleration of the center of mass? My intuition says internal, but the acceleration of the COM is clearly not $\large \frac{g(m_1+m_2)}{m_1+m_2}$ (I tried using $a_\text{COM}=\large \frac{F_{\text{net}}}{\text{total mass}}$).
What am I missing here?
Whether a force is internal or external depends on how we define the system that we are considering.
Tension is an external force if we are considering the motions of the masses separately (so each mass is a separate system) but tension is an internal force if we are considering the two masses (and the string) as one single system. If we consider the two masses as a single system, then the net force acting on it is $(m_1+m_2)g - N$. $T$ does not appear directly in this expression - but to find $N$ we need to find $T$ anyway.
So it is simpler in this case to consider the two masses separately to get two equations of motion, and then eliminate $T$ from these two equations.
Well, if you want to find the acceleration of the center of mass of the whole system, then you need to consider the whole system as one body. Hence, the tension forces between the masses and the pulley will be considered as internal forces.
However, the tension force holding the pulley to the ceiling is an external force as it is applied by a body that is not part of the system. Therefore, you need to consider it when evaluating $F_{\text{net}}$, as Dale said.
We know that there is no acceleration (it's acceptable to use intuition to make this kind of assumption in cases like this) so we find that the upwards tension force is equal and opposite to the force due to gravity. Alternatively, if you know the tension force you can add it to the gravitational force and find that $F_{net}=a=0$.
