Garret's presentation of what he calls the "Einstein-Podolsky-Rosen-Garret" paradox, in the 25:00 - 29:00 range of the video you link to, is not sound.
Garret proposes a source of entangled particles which produces the state
$$
\newcommand{\up}{|\!\uparrow⟩}\newcommand{\down}{|\!\downarrow⟩}
|\Psi⟩=\frac{\up\down+\down\up}{\sqrt{2}}
$$
and then sends the particles to distant locations, to be measured at spatially-separated events. His protocol then asks you to 'measure on the left, and look for interference on the right', which can be succinctly phrased as measuring on the $\{\up,\down\}$ basis on the first mode, and on the
$$
\newcommand{\plus}{|+⟩}\newcommand{\minus}{|-⟩}
\left\{\plus=\frac{\up+\down}{\sqrt{2}},\minus=\frac{\up-\down}{\sqrt{2}}\right\}
$$
basis on the second mode. The measurement probabilities in this case are easily seen to be
$$
\newcommand{\bup}{⟨\uparrow\!|}\newcommand{\bdown}{⟨\downarrow\!|}
\newcommand{\bplus}{⟨+|}\newcommand{\bminus}{⟨-|}\newcommand{\bpm}{⟨\pm|}
\left|\bup\bpm|\Psi⟩\right|^2\frac12,
$$
and analogously for $\bdown$, so if you measure on the left there is no interference on the right.
Garret then claims that this can be used for superluminal communication, and this is where he is incorrect.
So far, there's nothing that Alice, who is in control of the first mode, can do to alter the outcome at all - she definitely cannot control which of the two outputs ($\up$ or $\down$) she will get. The only choice she has is whether to measure her system prior to the interference step, or to let the two arms interfere and then measure. What Garret apparently doesn't realize is that even if Alice does let her system produce interference, the other system will not produce interference either. Alice has no way to make Bob's side of the system display interference without sending him classical information at subluminal speeds.
Let me sketch that calculation as it is important to the argument. Suppose both Alice and Bob measure on the $|\pm⟩$ basis, with Alice obtaining $|a⟩=|\pm⟩$ and Bob obtaining $|b⟩=|\pm⟩$. The probability for this outcome is then
\begin{align}
|⟨a|⟨b|\Psi⟩|^2
&=\frac18\left|(\bup+a\bdown)(\bup+b\bdown)(\up\down+\down\up\right|^2
\\&= \frac18\left|a+b\right|^2.
\end{align}
Thus if Alice gets $\plus$ it is certain that Bob will get $\plus$, and ditto for $\minus$, so it seems that Bob does observe interference. However, just because Alice decides that she wants to run her system through the 'recombine' step that doesn't mean that she gets to control which outcome she gets. She will get $\plus$ as often as she does $\down$, which means that so will Bob, and what that looks like to Bob is simply no interference.
Alice's actions, then, have no effect on what Bob observes, and therefore this channel cannot be used for superluminal communication.
This does sort of have a bearing on the Copenhagen Interpretation as described by Garret,
most people think of the Copenhagen interpretation as the idea that measurement causes a physical phenomenon known as “collapse of the wave function” which is non-linear and irreversible, i.e. it causes a quantum superposition to change into a probabilistic mixture of classical states, and that there is an physical difference between these two states.
but it doesn't rule CI out. In particular, this understanding of the Copenhagen Interpretation does force Bob's system to change instantaneously as soon as Alice measures her side of their entangled pair, but Bob's system somehow contrives to (partially) "hide" this inner change of state from any possible measurement, in a way which exactly prohibits superluminal communication.
This bothers some people (it somehow imagines Nature as having an extra set of supernatural powers which it purposefully denies us) but it is not inconsistent with the laws of physics. In particular, this "EPRG" argument does not rule out the Copenhagen Interpretation as "scientifically untenable", as Garret appears to claim.
I'm afraid I won't have time to critique whatever it is he says in the second half of the video, though. Quantum interpretations are a tricky business, and if you make technical mistakes on the underlying mechanics then there's no telling how much of a basis the resulting arguments will have.
Best Answer
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.