You can trivially apply MWI to QFT, so just do it and ignore nay sayers.
Or if you want to talk to them, then the burden is on them to argue why you can't do something that you are literally doing.
After all, it is just too hard to predict what imaginary concerns they have.
Now, in your case a specific concern was listed:
Because in QFT time is relative, and running branching or collapses in different frames turns out to be inconsistent.
But the point of MWI is that branches fork and and that at some point it becomes practical for the each forked branch to lapse into a kind of solipsism and regard itself as like an entire self contained world. But it also doesn't have to do so, so it doesn't matter when or if that thermodynamic limit exerts itself as justifying the solipsism. Plus, any opinions about frames would be about a branch examining itself.
This is because information that you can use probability about is information about a branch and the properties of a branch.
In fact, when I search for information on this I find the complete opposite, that it can be trivially extended to QFT with absolutely no problems.
It's just about linearity. A linear theory can allow different worlds as branches because they can act as if they were the only world.
The responder and I have differed before when it comes to determinism/randomness, and I do not know whether some of it may be ideological
Your quote objected to deterministic and indeterministic theories, so that's not it. This is about an imagined problem between branches and some imagined global absolute time. And a confusion about the entire point of MWI which is that everything is in that branch, itsit's called a world because it actscan act as if it were everything.
And really time symmetry should be the objection. Retrodictions about the past for a branch would be wrong if it actually thought it was the whole world.