What is the Copenhagen interpretation of quantum field theory?

The Born rule (and hence any discussion of collapse in the sense of the Copenhagen interpretation) is relevant only when an observer has made a distinction between a (tiny, observed) system and its (huge, observing) environment (= everything else, containing in particular the measurement equipment).

This distinction (not present in relativistic QFT itself) already breaks Lorentz invariance. The collapse (describing conditional probabilities conditioned on observations) is a property not of the wave functional in QFT but of its restriction to the Hilbert space of the observed system, which is an observer-dependent, vanishingly small part of the Hilbert space of the complete (observed + measuring) system.

This restricted few particle system is only an effective theory, to which fundamental considerations cannot be applied.

Thus there is no contradiction. A sequence of papers with the title Classical interventions in quantum systems by Asher Peres discuss how observations by different observers remain consistent in a relativistic framework.

I take a minimal interpretation of QFT in a Copenhagen style to seek to make a connection between a classical description of/model for an experimental apparatus and classical records of its measurement results and a QFT model for the same apparatus.

Classically, a modern measurement device is most often a thermodynamically metastable system that we engineer to make transitions from a Ready state to a Detected state, and for which we also engineer an explicit feedback that brings the state back to Ready as soon as possible. For such a device, electronics detect a change of voltage from 0V to 1V, say, and makes a classical record of the approximate time at which the transition happened (and, perhaps, of various classical settings of the apparatus at the time; see Weihs's Bell experiment for a concrete fairly straightforward example, http://arxiv.org/abs/quant-ph/9810080). Typically we make millions of such classical records, which we group together in some way or another to construct ensembles (for Weihs, two events at close enough to the same time = one element of the highest-level ensemble, which can be split into 16 sub-ensembles according to the recorded classical settings). From this, we can construct statistics and show that they correspond or do not correspond well to whatever QFT models we may have constructed for the experiment (for the simplest cases, QFT is pretty much just quantum optics, we don't have to worry much about the interacting QFT of the later-added part of your question, and the asymptotic fields associated with S-matrix results are about as straightforward).

There is a classical more-or-less continuous signal that underlies the discrete events, which hardware and software converts into times when thermodynamic transitions happened (for the sake of storage limitations because recording the signal picosecond-by-picosecond would be enormous and likely irrelevant). The signal is rather imprecise, in that it's not an observable quantum field along a time-like trajectory, which is not possible because of the field commutation relations, but is instead a functional of thermodynamically large numbers of DoFs associated with the measurement device, for which field commutation relations have far less effect than the change from 0V to 1V that signals a measurement event. Nonetheless, we take it that the statistics of events are coupled to the rest of the experimental apparatus and will be changed by any change to the rest of the experimental apparatus. Whatever changes there are to the recorded statistics can be modeled by choosing a different state of the quantum field (or alternatively by choosing a different operator). For a given measurement operator, we can perhaps reasonably say that the state of the quantum field "causes" the observed statistics to be what they are (rather close though this is to commonly discounted ensemble interpretations of QM, https://en.wikipedia.org/wiki/Ensemble_interpretation), but perhaps it's as well to be more reserved when choosing whether to claim that the quantum field causes individual observed events.

From this point of view, the "collapse" is a classical property of an experimental device that we have engineered to be in a thermodynamically metastable state. If one also takes the view that QFT is an effective field theory that is essentially stochastic, the Lorentzian dynamics are a property of the statistical-macroscopic level of model, so that we cannot make any direct claim about the symmetries of the dynamics at the level of individual events. Indeed, we know that the macroscopic effective dynamics of superfluid Helium, being Lorentzian but with the speed of sound replacing the speed of light, is significantly different from the microscopic dynamics, so we should not rush to assume that the dynamics associated with individual events has the same symmetries as the dynamics associated with the statistical level dynamics. This is not to claim that there is a particular FTL deterministic dynamics, potentially associated with a different metric as in the superfluid Helium case, but it leaves the door open for one, which is all I feel the need for because I'm mostly content just to use QFT; if you want a specific choice of dynamics at the level of individual events, that's harder. Current experiments are very far from ruling out all possible classical local dynamics, they can only rule out the straw man of Lorentzian dynamics.

Perhaps we can also reasonably note that modern Quantum Gravity approaches give up Lorentzian dynamics at Planck scales with the intention that we will be able to show that the effective dynamics at large scales will nonetheless be Lorentzian.

You will note that the above doesn't much engage with GRW as it is usually described, for which collapse is not nearly so tied to experimental details as I have it above, which I suggest is more as a Copenhagen-style interpretation should have it. The distinction between stochastic/statistic and deterministic levels of description is of course problematic in its evocation of Einstein's later worries about quantum theory, which I suggest, however, can be revisited with modern ideas about effective field theories in mind, if one cares enough and can think of a way to do it.