[T]here are many classical worlds equally real to the single world we experience. Surely you accept this.

Surely? I don't even know what it means. Words "real" and "experience" are close neighbours in my vocabulary, real unexperienced world sounds a lot like an oxymoron, at least if not based on a really strong argument.

What gets worse when you move to QFT?

Nothing. I have tried to (incompletely of course) explain the relation between the conventional and relativistic Hamiltonian formalism in case of mechanics, where it is slightly more intuitive and simpler. If you address my first remark, you have disinterpreted it. I don't say that move to QFT isn't justified, but that one conventional argument used to support this move isn't good.

The classical states are not ontologically fundamental in ordinary QM. If that's what you mean by MWI...

It isn't. By MWI I mean probably the same thing as anybody else. Nothing particularly related to classical states. I have discussed classical states in order to give some background to my intuitions. My statement was that probably the quantum states are a redundant concept.

I'm happy to talk about QFT as a bunch of ordinary QM systems, one for each reference frame.

I would understand that QFT is a bunch of QM systems, one for each spacetime point. I don't understand what reference frames do with it. In any fixed reference frame QFT has infinite number of degrees of freedom. Maybe you speak about momentum representation? I am confused.