First thing to note, possible worlds can't be specified at different levels of detail.

Let's take possible worlds to be sets of propositions and truth values.

I think that these two desires are contradictory. Part of what I'm trying to say is that it's a highly nontrivial problem which propositions are even meaningful, let alone true, if you specify possible worlds at a sufficiently high level of detail. For example, at an extremely high level of detail, you might specify a possible world by specifying a set of laws of physics together with an initial condition for the universe. This kind of specification of a possible world doesn't automatically allow you to interpret intuitive referents like "I," so the meaning of a statement like "I am holding a glass of water" is extremely unclear.

you have raised concerns for practical issues (how we specify worlds) and epistemic issues (how we know what's in worlds) but these aren't really relevant to the issue of rigid designation.

How do you know what things are rigid designators if you neither know how to specify possible worlds nor how to determine what's in them?