The pulses intersect at the mid point and, for a while, a standing wave is formed by their superposition, and in that standing wave are a series of stationary points at which the E and H fields are 0.
Unfortunately, your assumption about how this works is incorrect. It is best to work through the math on problems like this. For clarity I will use natural units ($c=\mu_0=\epsilon_0=1$) and a frequency such that $\omega=k=1$. Let $$\vec E_1 = (0,A,0) \cos(x-t)$$ $$\vec B_1=(0,0,A) \cos(x-t)$$ and let $$\vec E_2=(0,A,0) \cos(-x-t)$$ $$\vec B_2 = (0,0,-A) \cos(-x-t)$$ and let $$\vec E_3 = \vec E_1 + \vec E_2$$ $$\vec B_3 = \vec B_1 + \vec B_2$$
So the first fields represent a plane wave traveling in the $+x$ direction, the second fields represent a plane wave traveling in the $-x$ direction, and the third fields represent the standing wave formed by their sum. I leave it as an exercise to show that all three pairs of fields satisfy Maxwell's equations.
Note that by simplifying we obtain $$\vec E_3 = (0,2A,0) \cos(t) \cos(x)$$ $$\vec B_3 = (0,0,2A)\sin(t) \sin(x)$$ So although it is correct that $\vec E_3$ has a series of stationary points at which it is always 0 (nodes) and it is also correct that $\vec B_3$ has a series of nodes, the nodes for $\vec E_3$ are different from the nodes for $\vec B_3$. There are no points that are always zero for both fields and there are also no times when both fields are zero everywhere.
To understand the flow of energy in this field configuration we can calculate the energy density and the Poynting vector. $$\vec S_3 = \vec E_3 \times \vec B_3 = (A^2,0,0) \sin(2t) \sin(2x)$$ $$u_3=\frac{1}{2}(E_3^2+B_3^2)=A^2\left(\cos^2(t-x)+\cos^2(t+x)\right)$$
However, both pulses are carrying energy, in the E and H fields, and that energy arrives at the final destination, so energy is being transferred, in the form of E and H fields, through points that never have an E or H field.
This is partly correct and partly incorrect. There are points that are nodes for the E field and different points that are nodes of the B field. There are no points are nodes for both the E and B field.
However, there are indeed nodes for the Poynting vector $\vec S_3$. The nodes of $\vec S_3$ include all of the nodes of $\vec E_3$ and of $\vec B_3$. Any point where $\vec S = \vec 0$ means that electromagnetic energy is not flowing through that point. So electromagnetic energy never flows through the nodes of $\vec S_3$ at any time. Instead, by looking at $u_3$ and $\vec S_3$ we see that EM energy merely oscillates back and forth between the nodes. There is no transfer of energy through the nodes of $\vec S_3$.
Just because energy arrives at the final destination does not imply that energy goes through any of the nodes of $\vec S_3$. Energy doesn't get some cosmic ID tag, so there is no sense in which you can say that the energy leaving one laser must be the same energy as the energy arriving at the other laser and therefore no justification for the assumption that energy must flow through the nodes. Energy is conserved, and energy conservation does not require it to flow through the nodes.
