Replacing "He" in the antecedent by "one of two boys" would be ambiguous, because mere repitition could mean the other boy. "the/this/that one of the two boys" makes a difference, to begin with. Therefore the question seems meaningful.
Representing anaphora with dependent types might not be the reference you want, but the one that you need.
I did not read it, just stumbled about it when trying to read up about dependent types w.r.t. this question. The linked page, ncatlab.org is dedicated to the field of category theory in mathematics, with a strong leaning towards theoretical physics. So "semantics" and "syntax" has a slightly different bent in this context, but if applied to Linguistics they [who?] hopefully try to marry the ideas from both families. I can't give it a proper introduction because I'm not a practitioner.
After thinking it through, your example reminded me of what dependent types seem to be. It's pure coincidence that the first search result talks about anaphora.
I'd be surprised if they talk about "specific indefinites" explicitly.
An introduction to universal algebra, type theory or similar theory might be a good basis (cp mention of "Homotopy type theory" and "monad" at dependent type theoretic methods in natural language semantics.
Dependent types have been implemented in various new programming languages, but it's a niche thing with many competitors around the block; It can be used as a Higher Order Logic (HOL), but it's not clear whether that's needed for natural language, or for this example. The genre of Liars Paradoxons might be relevant here, if i.e. "The brother who shaves all brothers who don't shave themselves" is undecidable in the chosen version of semantics, which seems to be avoided in dependent linear type theory due to the absence of diagonalizations.
Simply typed lambda calculus for example is a theory closer to the field of syntax (linguistic). However, given a syntax-semantics duality, a concept in syntax theory should have a dual in semantics. If you will, one could say that the PoS of "He" in the anaphora is underspecified, because "... who should ..." would work as well; It's not clear whether that's a relative, interrogative or personal pronoun, in my humble opinion.
The idea becomes clearer if rephrasing the example:
One of the brothers must clean the room.
It should be done by friday.
I would argue naively, that an indefinite "it" often refers to the whole abstract content of the preceeding message. It is a reference to an object, but the object is a function that may be reevaluated until it yields an atomic result. This would be realized by a maybe-monad, so called promisses in Functional Reactive Programming.
There is, after all, no common expectation that the Brothers couldn't do it together although it may be discouraged or impossible. The text leaves open how the difference is to be resolved, especially the delegation of the task. In that sense, the "one of the brothers" is not indefinite at all; Quite the opposite is the case, if the responsibility was laid onto both, or on an overseer to the brothers, potentially including the chance to find an agreement and negotiate the task between them. That's a question of discourse analysis ("protocol" as I like to call it).
The use of "should" is also notable. "I would not do that, if I were you" is very indefinite, too. In contrast, "It puts the lotion on its skin" has a grating edge and "He completes the task by friday" would seem wrong, too confident, doesn't it? It would sound like a narration of completed events in a present tense voice, not like a desiderative. So you really do not get around a look at the syntax.