"Rest in peace" does not function as a verb in this sentence. It's simply a sentiment, like "hello" or "my thanks." Regardless, that does not affect the state of the object phrase.

You are correct that the example sentence is a clear parallel for the one we are discussing, but you are still making the same mistake in it. That sentence would read as follows:

> Give the recipe to [whoever has to cook the meal].

If you look at the bracketed phrase, it is clear that the subject of the phrase must be in the subjective case. You can't say "him has to cook the meal." If you put the pronoun in the objective case to satisfy the preposition, you are robbing the following phrase of its subject. If you give the "has to cook" phrase its subject, then the entire phrase works, and in its whole functions as the object.

The reason that looks strange is because in this construction, we are actually omitting a word. The proper way to say this sentence would be like so:

> Give the recipe to [the person] who has to cook the meal.

When it is worded as such, you can see that "the person" functions as the object of the initial phrase, and "who" functions as the subject of the second phrase. That second phrase is actually a dependent clause, so it requires a subject.

However, when you omit the object ("the person") of the first phrase, then the entire second clause becomes the object. It still, however, remains a complete clause with its own subject—and a pronoun in objective case cannot function as a subject.

This link does a good job of explaining our dilemma, using examples like the initial sentence, and the one you provided.