The question is, should the 'should' in this sentence be capitalized?
Also, is the type of that sentence interrogative or declarative or a mix of both?
I invariably struggle towards the simplest possible effective, clear, unambiguous answer. Often, these goals can't be fully met, but I try.
In my opinion, the best version of your sentence is this:
The question is, "Should 'should' in this sentence be capitalized?".
The virtue of this version is that it is absolutely unambiguous. Clarity in communication is usually my number one goal.
The sentence is declarative. An interrogative sentence is embedded within this declarative sentence as nothing more than the object of the sentence. It does not change the nature or kind of the sentence any more than dog changes he petted his dog into any other kind of sentence. It is possible to omit the concluding period as being unnecessary punctuation, and I wouldn't quibble over that, but even if you took it out, it would not change the sentence into an interrogative. The mere fact of the concluding punctuation (in that case) being a question mark would not define what type of sentence it is, because it applies only to the embedded question.
I accept, however, that it's possible to reduce the complexity of your sentence down to this:
The question is, should the "should" in this sentence be capitalized.
In this case, the embedded question is reasonably well understood from "should" beginning the object phrase, and the question mark is not needed. The period should be used by itself then. I think this version has the virtue of simplicity, although some degree of ambiguity starts to sneak in, so I would be wary of that.
I would not agree with a version that removes the comma because then you would create syntactical confusion. The reader would read the sentence as "The question is should," and would be confused about where to go from there; "is should" would seem to be an impossible verb phrase, and suddenly the reader would be lost.
As for your choice of whether to capitalize the word when you are in fact asking whether to capitalize it or not, do it either way, and see what answers you get. Or be creatively clever, thus:
The question is, Should I be considered correct in capitalizing "should" to start this sentence?
The main clause is declarative:
A. The question is <something>.
What looks like an embedded clause is interrogative:
B. Should the 'should' in this sentence be capitalized?
It looks like you're embedding B into A, but the embedded clause doesn't take the form of a subordinate clause. If it did take that form, you'd write it like this:
The question is [ whether the 'should' in this sentence should be capitalized. ]
So it seems instead that B is used like a quote. There's no grammatical requirement for quotation marks, though. There are plenty of ways to write this:
1. The question is, "Should the 'should' in this sentence be capitalized?"
2. The question is: Should the 'should' in this sentence be capitalized?
3. The question is: should the 'should' in this sentence be capitalized?
4. The question is, should the 'should' in this sentence be capitalized?
You do need something separating the two, though, because otherwise the reader expects a subordinate clause:
5. *The question is should the 'should' in this sentence be capitalized?
Although it's arguable that this is merely a stylistic choice, in which case you might consider example 5 acceptable.
The actual question about whether should should be capitalized is definitely one of style. The way you've written it, I think the answer is no.
The Cambridge Grammar of the English Language describes this construction in chapter 20, section 3.2.4. First, it describes a related reported speech construction:
In this construction the reported speech is a complement of the reporting verb ... The direct reported speech complement (whether enclosed in quotation marks or not) is required to be preceded by a punctuation mark, usually a comma. A colon is also possible ... (emphasis added)
Then the construction we're talking about here is described more directly:
The question to be considered next is: "How long should artificial respiration be continued in the absence of signs of recovery?"
In [this example] the colon occurs before the complement of be in its identifying sense -- a complement, moreover, which has the form of a main clause; this case has affinities with the reported speech construction. [These constructions] are exceptional: in general, a verb may not be separated from its complement by punctuation.
So this type of sentence is fairly unusual, but it's nonetheless standard to put the comma between the verb and its complement, and it's also okay for the complement to take the form of a main clause.
In this answer, the * symbol marks a sentence as unacceptable in standard English.
There are nearly 300,000 written instances of the text the question is should in Google Books, but if you leaf through a few pages you'll see there's very little consistency about how they're punctuated.
Some writers include no punctuation. Some put a comma, colon (and/or hyphen), or even semicolon before the "embedded question text" (i.e. - after the word "is"). Some put single or double quotes round the embedded question (in which case they invariably capitalise "Should"). Others capitalise it even if they don't use quote marks. And they usually end with a question mark.
In short, this is a stylistic choice. Personally, noting that the general tendency is to use less punctuation than might have been common in the past, I would either elect to use none at all, or at most maybe just put a comma after "is" and/or append "?". I certainly wouldn't capitalise - it looks "old-fashioned" to me.
It's interesting to note that in (case-sensitive) NGrams, the capitalised version doesn't even occur often enough to be graphed. Grammatically it makes no difference whether the embedded question starts with should or with, say, what. But people are strongly conditioned to recognise sentences (or sub-elements within sentences) starting with wh- words as wh- questions, so they capitalise a bit more with "what".
In my opinion, it's not particularly useful to classify the utterance as either interrogative or declarative. It can be seen as either (the question mark makes it "interrogative", and it's often uttered as a question, but structurally it's obviously a "declarative" statement).
If you don't want to use quotes, I would opt for a different approach:
When asking how, I usually see "The question is how the dog got there." rather than "The question is, how did the dog get there?"
If we apply this approach to should, it becomes "The question is whether or not 'there' is where the dog should be."
I think that would the best method, as it removes the necessity for embedding. However, it is a significant change in a matter of stylistic choice, so do whatever you think is best.