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Are there extensions of predicate logic which focus on resource models, possibly even in actor models?
And if so, would it be possible to reason on a meta-level on how many resources should be expended to find further proofs in the same or other systems?
Do you mean a resource-aware logic? If so, yes, there's linear logic [Girard, 1987] (as was already noted in the comments). It has had a big influence on the study of concurrency and implicit computational complexity. You might be interested in the latter, where reasoning is focused on the efficiency of programs.
What do you mean exactly by "actor models"? If you mean concurrency, please have a look at: What is the Curry-Howard analogue for linear logics? If you mean identifying who/what uses/implements resources at the level of propositions, we explored an extension of linear logic for that [Carbone et al., 2016], but mainly for the notion of communication protocol (who performs which communications). Useful inspiration might also come from hybrid logics that support notions of "worlds", like this one: https://www.cs.cmu.edu/~rwh/theses/reed.pdf
