I think there is no answer to this as simple as you're hoping for. My intuition is that the external force might actually be lowered by something similar to what happens in the Venturi effect (https://en.wikipedia.org/wiki/Venturi_effect).
Let's look into this, and first add some assumptions. Let's presume that at the openings you marked with $F_2$ and $F_3$ there are pipes attached whose outflow is controlled by valves that we can close, and that the influx into the container is equal to the outflux at those valves such that the water level in the container remains constant. Let's further assume that the pressure at the free surface in the container is constant (atmospheric pressure, for example), that the fluid is incompressible and that we can neglect viscous effects in the volume element you are interested in.
Now, you can follow the following with Landau & Lifshitz, Volume 6, I'll refer to the equations therein. We use Bernoulli's equation for an incompressible fluid (10,4 in L&L)
$$
\frac{v^2}{2} + \frac{p}{\rho} + g z = \text{const.}
$$
wherein $v$ is fluid velocity, $p$ pressure, $\rho$ density, $g$ gravitational acceleration and $z$ height in some cartesian coordinate system. Compare the pressures $p_1$ and $p_2$ at one specific point of the plate for two different velocities $0 = v_1 < v_2$ and you'll find that the pressure $p_1$ when there is no fluid flow is higher. I.e. switching on the fluid flow, say by opening the valves I wished for, we actually lower the pressure at the plate and thus the external force. Which we can calculate directly by integrating the force density
$$
p \vec{n} + \rho \vec{v}\left(\vec{v}\cdot\vec{n}\right)
$$
(7,4 in L&L) over the plate. $\vec{n}$ denotes the normal vector to the plate. And by a no-flux boundary condition $\vec{v}\cdot\vec{n}=0$ and we only have to integrate the pressure.
You could actually try and find an analytical solution for a potential flow in the given situation (look for stagnation flow, you'll find something on this in chapter 10 of L&L).
Unfortunately, if the dimensions of the system or the flow rates are such that viscous effects play a significant role, the flow becomes turbulent, the above explanation doesn't work and I wouldn't know how to answer your question...
