Harlow's equation (3.34) is a schematic way of expressing the idea that the gradient term in the Hamiltonian contributes $\sim 1/\epsilon^4$ to the expectation value of the energy density at the L/R interface when the state is a product state. I'll explain this in detail using lattice QFT, which implements an explicit version of the "UV length cutoff" that Harlow calls $\epsilon$.
Definition of the model
The Hamiltonian for the free scalar field with mass $m$ is
$$
\newcommand{\bfu}{\mathbf{u}}
\newcommand{\bfx}{\mathbf{x}}
\newcommand{\half}{\frac{1}{2}}
H = \epsilon^3\sum_\bfx H_\bfx + \text{const}
$$
where $H_\bfx$ is the energy density operator
\begin{align}
H_\bfx &= K_\bfx + V_\bfx
\\
K_\bfx &= -\half
\left(\frac{1}{\epsilon^{3}}\frac{\partial}{\partial\phi_\bfx}\right)^2
+\half m^2\phi_\bfx^2
\\
V_\bfx &= \half\sum_{\bfu}
\frac{(\phi_{\bfx+\bfu}-\phi_\bfx)^2}{\epsilon^2}
\tag{1}
\end{align}
where the index $\bfx$ runs over lattice sites, $\bfu$ runs over a set of three basis vectors for the lattice, and $\epsilon$ is the distance between neighboring sites. The operator $\sim -(i/\epsilon^3)\partial/\partial\phi$ is the lattice version of $\dot\phi$, the canonical conjugate of $\phi$, and the quantity $V_\bfx$ is the lattice version of the spatial gradient term $\sim(\nabla\phi)^2$.
A state is represented by a normalizable (square-integrable) complex-valued function of the field variables: $\Psi[\phi]$. The inner product of two states is defined by integrating over all of the field variables:
$$
\langle\Psi_1|\Psi_2\rangle
\equiv
\int \prod_\bfx (\epsilon\,d\phi_\bfx)\ \Psi_1^*[\phi]\Psi_2[\phi].
\tag{2}
$$
A factor of $\epsilon$ is included with each factor of $d\phi_\bfx$ in the measure because the field varaibles $\phi_\bfx$ in (1) have the same units as $1/\epsilon$, but we want the inner product to be dimensionless. We can get rid of those factors of $\epsilon$ in the inner product by switching to a normalization in which the field variables $\phi_\bfx$ are dimensionless. Then equations (1) are replaced by
\begin{align}
H_\bfx &= K_\bfx + V_\bfx
\\
K_\bfx &= -\half
\left(\frac{1}{\epsilon^{2}}\frac{\partial}{\partial\phi_\bfx}\right)^2
+\half \frac{m^2}{\epsilon^2}\phi_\bfx^2
\\
V_\bfx &= \half\sum_{\bfu}
\frac{(\phi_{\bfx+\bfu}-\phi_\bfx)^2}{\epsilon^4}
\tag{3}
\end{align}
and the inner product (2) is replaced by
$$
\langle\Psi_1|\Psi_2\rangle
\equiv
\int \prod_\bfx d\phi_\bfx\ \Psi_1^*[\phi]\Psi_2[\phi].
\tag{4}
$$
The vacuum state
The vacuum state $\Psi_0[\phi]$ is the state that minimizes the expectation value of the Hamiltonain $H$. The vacuum state can be determined explicitly in this model, as shown in the body of this question, which however uses continuous space and $m=0$. Here, instead of writing down the explicit expression, I'll use some intuition to deduce its key properties.
Temporarily suppose $V_\bfx=0$. Then the Hamiltonian $H$ would describe a set of uncoupled harmonic oscillators, one per lattice site $\bfx$. The ground state would be the product of the individual harmonic oscillator ground states, each a function of $\phi_\bfx$ for just one lattice site.
The term $V_\bfx$ describes interactions among those harmonic oscillators. To find the vacuum state of the full Hamiltonian, including the $V_\bfx$ term, we need to balance the expectation value of the $V_\bfx$ term with the other terms. To do this, we need to replace the product state with a state that drops rapidly to zero whenever the $\phi_\bfx$s at neighboring sites differ much from each other, so that the numerator $(\phi_{\bfx+\bfu}-\phi_\bfx)^2$ of the gradient term is always small. (In other words, we need all of the oscillators to be entangled with each other.) The state that achieves this balance — the state that minimizes the total energy when the interaction term $V_\bfx$ is included — is the vacuum state.
From now on, suppose that the constant term in (1) has been chosen so that the energy of the vacuum state is zero.
Energy density in a product state
Now, following Harlow, divide space into two halves L/R and consider a product state
$$
\Psi[\phi]=\Psi_L[\phi]_L \Psi_R[\phi]_R
\tag{5}
$$
where $[\phi]_L$ means all of the $\phi_\bfx$s with $x\in L$, and similarly for $[\phi]_R$. The vacuum state cannot be written in this form, because a state of this form cannot drop rapidly to zero whenever the $\phi_\bfx$s at neighboring sites differ much from each other. For a product state (5), in the integrand of the expectation value, nothing constrains the numerator $(\phi_{\bfx+\bfu}-\phi_\bfx)^2$ of the gradient term to be small at the L/R interface. Equation (3) therefore says that the expectation value of that term is therefore of order $1/\epsilon^4$ at the interface when the state has the form (5), exactly as Harlow claimed. This shows that a product state is nothing like the vacuum state, at least not near the interface. The product state necessarily has a firewall.
The meaning of Harlow's equation
Now we can see what Harlow really meant by
$$
\partial_x\phi\big|_{x=0}\propto\frac{1}{\epsilon^2}.
\tag{3.34}
$$
This "equation" is a schematic way of conveying the idea that the quantity $\phi_{\bfx+\bfu}-\phi_\bfx$ in the numerator of the gradient term is not small at the L/R interface when the state $\Psi$ is a product state, as I explained above. Harlow is using the normalization that I used in equation (1), so that $\partial\phi$ has dimensions $1/\epsilon^2$, and (3.34) is understood to be a statement about the typical numeric value of this quantity in the integrand of the expectation value, where the $\phi$s are integration variables.
