Energy conservation for the beam splitter from first principles

Let’s consider a (cubic) beam splitter with two electric input fields $E_1,E_2$ and two electric output fields $E_3,E_4$ as illustrated in the figure below.

For a lossless beam splitter, we almost always find the literature claim
$$
vertboldsymbol{E}_1vert^2+vertboldsymbol{E}_2vert^2
=
vertboldsymbol{E}_3vert^2+vertboldsymbol{E}_4vert^2
tag{1}
$$
which intuitively matches our idea of a lossless beam splitter.

However, assume we don’t take eq. (1) for granted are highly skeptical of our intuition. How can we derive eq. (1) from first principles?

Strategy A: energy density

In strategy A, we take the definition of the electromagnetic field’s energy density,
$$
u
=
frac{epsilon_0}{2}boldsymbol{E}^2+frac{epsilon_0c^2}{2}boldsymbol{B}^2
tag{2},
$$
for granted.
Using $c^2boldsymbol{B}^2=boldsymbol{E}^2$ for the free field solutions of Maxwell’s equations, the energy density becomes
$$
u=epsilon_0boldsymbol{E}^2
tag{3}.
$$
So far, so good, the total energy of the field is obtained by integrating over the total volume,
$$
H
=
int dV u
=
epsilon_0int dV boldsymbol{E}^2
tag{4}.
$$
However, eq. (4) is only valid for a single field. I feel uncomfortable to state
$$
H_1+H_2
=
H_3+H_4
tag{5}
$$
as we do not know what happens in the beam splitter volume (so far, the beam splitter is a black box).

Strategy B: energy flux

Because of the afore problem, I tried another approach that uses the energy flux (or Poynting vector)
$$
boldsymbol{S}
=
epsilon_0c^2boldsymbol{E}timesboldsymbol{B}
tag{6}.
$$
For freely propagating light, we can use $cboldsymbol{B}=boldsymbol{e}_ktimesboldsymbol{E}$ wherein $boldsymbol{e}_k$ is the unit vector pointing in the direction of propagation. We then find
$$
boldsymbol{S}
=
epsilon_0cboldsymbol{E}^2boldsymbol{e}_k
tag{7}
$$
to be the energy flux of a single electromagnetic wave propagating in the direction of $boldsymbol{e}_k$.

Now, let’s define the $x$ axis to point in the direction of $boldsymbol{E}_1$ and the $y$ axis to point in parallel w.r.t. $boldsymbol{E}_2$.
Furthermore, we say that the energy flux at the deflection point in the center of the beam splitter has to vanish, i.e.,
$$
boldsymbol{S}_1+boldsymbol{S}_2+boldsymbol{S}_3+boldsymbol{S}_4=0
tag{8}.
$$
Obviously, eq. (8) cannot be true because $boldsymbol{S}_2$ and $boldsymbol{S}_4$ are orthogonal to $boldsymbol{S}_1$ and $boldsymbol{S}_3$ and this would lead to $boldsymbol{E}_1^2=boldsymbol{E}_3^2$ and $boldsymbol{E}_2^2=boldsymbol{E}_4^2$.

What am I missing?