There is no structural reason to have two determiners, so I believe this can be treated as definiteness agreement. In other words, the underlying structure looks roughly like English*, then the additional forms are via agreement (or some kind of copying) rather than phrase structure.
Arguably there is still some more complicated morphological structure, but that would (probably) be within the heads of the phrases, "below" the syntax.
(*It often does, doesn't it? That's what happens when a theory is developed for English then extended for use with other languages. I'm skeptical of this and not necessarily endorsing the approach, but if you follow the basic assumptions in Carnie, you will reach a conclusion along these lines. Whether, starting from Hebrew, you'd make those assumptions is another question.)
I don't know of specific work that deals with this, but note that the same phenomenon is found (almost identically) in Arabic, so that's another place to look for research if you're interested.
I'd recommend this article (Kramer 2010) on Amharic, a related Semitic language:
http://dx.doi.org/10.1111/j.1467-9612.2010.00139.xIn Amharic, there are some complicated (but interesting) distributional facts about the definite article. It always appears in second position, after whatever the first constituent is within the NP. Within (fairly complicated) Minimalism, Kramer argues that this is due to linear displacement: it's actually at the beginning, but phonologically must attach to the end of a constituent. So it just jumps over to the end of the first thing it can. That may seem a bit strange, but the explanation works out well.
Beyond this, additional definite markers sometimes appear in the NP. Unlike in Arabic and Hebrew, however, these are optional in Amharic. So you can tell which is the "real" definite marker, and there is strong evidence for some sort of agreement effect to create the additional copies.
The same analysis would explain Arabic and Hebrew, if we ignore the 1) linear order [the article attaches to the beginning in Arabic and Hebrew] and 2) the optionality of the agreement-- it's optional in Amharic but not in Arabic or Hebrew.
So you may or may not want to get into this level of analysis with other languages yet. Carnie is walking you through the theory slowly and you're not yet ready (from the book, that is) to analyze certain phenomena. Beyond this, Generative syntactic theory does not actually explain linear order in some cases. That may be due to (phonological) displacement or other effects, post syntactic. Regardless, if it is to be explained, it'll be closer to the chapter on movement and so forth.
I hope that helps with an overview. Your thoughts seem about right for the tree and so forth, assuming that it made sense to represent it as part of the phrase structure.
I’m not completely sure that this is “grammatical,” but I hear this form used all the time in colloquial Hebrew. I really don’t know if there is a more proper form to express definite comparative adjectives.
Take another look at the early chapters in Carnie

That's exactly what "grammatical" means-- that speakers use it, because it's part of their grammars. Whether something is or is not "proper" is irrelevant (that is, an unrelated question).