I think I may have (tried to) state this before, but it seems like there are two interpretations of the Benecerraf dilemma. A strong one and a weak one. The strong one can be called a 'indeterminacy reading' and the second a 'underdeterminacy reading'.
It seems like in some cases, like when someone shows me photos they claim were taken at a haunted house, and point to some glossy flares they call 'orbs', the person is claiming that some shaky evidence should convince us of something's objection. In a case like this, it seems indeterminate what those 'orb' things are. They could be anything, so they are nothing. That is, 'orb' is being applied to a tokening of a candidate for being an orb. But something seems funky about the identifying of the tokening as an orb.
Sometimes the Benacerraf dilemma seems to run like this. If you want to identify numbers with abstract entities, you have to say something about those entities. You seem to leave it indeterminate as to what they are. If you face multiple interpretations or multiple candidate objects, it seems indeterminate which one to choose.
But in some cases, like when a scientist sees a cloud of particles under his microscope, he simply has evidence that is underdeterminate. He clearly has some candidates in mind, and some principled reasons to select some interpretations of candidate entities over others. Or, if a policemen found Jones dead, he might be sure that there is a murderer, even though the evidence underdetermines who that murderer is. In some sense I guess who the murderer is is indeterminate. But it seems wrong to think that anything could have killed Jones. It seems we can at least narrow it down to a someone.
It seems then that the lover of propositions only needs to appeal to common-sense intuitions that are consistent with the existence of propositions. What propositions are doesn't seem to be indeterminate, in the sense that we have no principled way of finding out what they are. We may simply have evidence which is underdetermined.
If I'm playing chess, for example, I can ponder at a •pawn• token. So I can have a wooden pawn, a metal pawn, a pawn shaped like a bear, a pawn shaped like a pillar, etc. There is something here, a shared structure of all the shapes I want to call pawnness, or a •pawn•. The 'pawn functional class'. Whatever. It seems odd to take the strong reading of the Benacerraf to claim that since anything could play the role of a pawn, that is, anything could be a pawn, nothing can be a pawn. It seems better to admit that the list of pawns is open ended, since it is underdetermined what pawns are in some sense. Anything could be used as a pawn!
It seems the same sort of move could be made with propositions. A shaky notion seems fine. We plead guilty to underdetermination. But this isn't the same as accepting indeterminacy. That many things might be pawns or propositions doesn't mean that nothing is.