A flexible spherical shell of charge orbiting an atom necessarily has an axis of rotation and an angular momentum and would thus be oblate. Furthermore, if we neglect quantum effects or prefer to think of them as not real, such a shell would have a continuous set of possible angular momenta, which is not observed. In fact, the ground state, say, of the hydrogen atom, has zero angular momentum and thus, from that point of view, is simply not equivalent to any sort of orbit whatsoever, or perhaps we prefer to think of it as a superposition of orbits which has the property that the angular momentum is zero - but the BLP ontology does not admit superpositions.

Furthermore, the properties of these sheets of electron material seem quite outlandish. Consider the ground state spherical state. If we force it to have zero angular momentum to match observation, then the shell must be rigid, but as you know, a rigid shell does not actually have a stable orbit around a central charge (this is kindergarten physics, feel free to work it out).

Things get worse when we consider high energy states which do have angular momenta. Does the nucleus get stick in one lobe or the other of the charged sheet, which is now infinitely thin (and thus dense) at the center?

This doesn't even begin to get into the myriad other reasons we use quantum mechanics and reluctantly give up our classical ontology besides just the description of atomic orbitals. To meet all these other demands (eg, Bell experiments) with the "sheet of charge" ontology is at least a herculean task which BLP's material does not accomplish (I have read it) and at worst has even greater foundational challenges than quantum mechanics. After all, we know "wave function collapse" (which I use here to refer simply to the totally uncontroversial measured phenomenon without an interpretation given) indeed happens "instantaneously" over spacetime - for a totally classical ontology this phenomenon is beyond peculiar and would indeed involve genuine signals propagating faster than light. At least in quantum mechanics we can understand wave function collapse in alternative ways which do not involve the violation of the laws of special relativity.

And this doesn't even get into how one would formulate QFT in the BLP picture. QFT isn't without its challenges, of course, but its a basic generalization of the structures of quantum mechanics. I don't know of any place where BLP calculates scattering amplitudes correctly, but feel free to show me where he works out ABC scalar field theory or whatever.

In the end the appeal of any theory needs to be evaluated in the full context of human knowledge, not just against a narrow set of objections about some qualities of a given theory. BLP seems motivated by a basic distaste for some of the weirdness of QM but evaluated in total, against the total sum of experiments routinely carried out in the world and against basic physical intuitions from even classical physics, it doesn't hold up.