Reconciling two types of time in QFT

The construction you described is, indeed, possible, but this is not the standard way to do QFT. In standard wave functional approach, we work in some specific (but arbitrary) reference frame. In this reference frame we chacterize basis field state as a function of space variables only: $\phi_x$. And in this reference frame we have unitary evolution $\exp(itH)$.

The theory constructed in this way contains only one time ($t$), but such theory is not explicitly Lorentz invariant. But in fact it is Lorentz invariant implicitly — if the Hamiltonian is correct — and this can be verified.

To construct an explicitly Lorentz invariant theory, other approaches are used. The approach that you described is called "parameterized QFT" and it is very rare. More commonly used approaches are second quantization and path integral quantization, and these approaches also contain only one time variable.

Let $M^4$ denote the four dimensional Minkowski spacetime. $$\Phi(x)$$ is the quantum field at the event $x\in M^4$ and here $x$ is nothing but a label.

Next there is the action of the (proper orthochronous) Poincaré group $\cal P$ in terms of isometries on $M^4$. If we fix arbitrarily a Minkowskian reference frame we can represent the Poincaré group with respect to these axes. Since the action of $\cal P$ on $M^4$ is transitive, the choice of a Minkowskian reference frame is irrelevant.

Let $p_0,p_1,p_2,p_3$ be the generators of spacetime translations along preferred Minkowskian axes. If $n$ is a timelike unit-vector future-pointing $n^\mu p_\mu$ is the generator of time displacements along $n$ (so a generic direction in the future light cone). The action in $M^4$ of the generated one-parameter group is $$\exp(\tau n^\mu p_\mu) x = x + \tau n$$ The action of the isometry group $\cal P$ can be implemented in the Hilbert space through a strongly continuous unitary representation $${\cal P} \ni g \mapsto U(g)$$ In particular whe have the representation of the subgroup $$U(\exp(\tau n^\mu p_\mu))= e^{-i \tau n^\mu P_\mu}\:,$$ where I have introduced the selfadjoint operator $n^\mu P_\mu$ representing (minus) the Hamiltonian operator associated to the temporal direction $n$. The action on the quantum fields is $$U(g) \Phi(x) U(g)^\dagger = \Phi(g(x))$$ In the special case of time translations, $$U(\exp(\tau n^\mu p_\mu)) \Phi(x) U(\exp(\tau n^\mu p_\mu))^\dagger = \Phi(x+ \tau n)\:.$$ You see that the parameter $\tau$ associated to the time quantum translation acts as as an isometry on the manifold: moving the labels of the points. Time evolution in Heisenberg picture is the inverse transformation of time trnaslation.

All that makes sense just because we have a coordinate system made of curves tangent to Killing fields of the manifolds. In curved spacetime, in general, this picture does not exist. In case a timelike Killing field exists a similar construction is still possible.

In general in QFT in Minkowski spacetime a safe point of view is Heisenberg picture, extending this picture to include the action of the full $\cal P$ not only time-translation (the inverse of time evolution).

EPR refers to correlations between regions of spacetime which are causally separated. So, we are dealing with "labels" and we are assuming that "$x$" and "$x'$" are such that there is no future-directed causal curve joining them. Here quantum time evolution does not enter the game. What happens is that outcomes of quantum measurments at $x$ and $x'$ on an entagled system turn out to be show correlations (...).