Learn English – Indirect questions using “do you think”

The syntax you want to use doesn't sound natural to me (also a native speaker) for an indirect question. What you seem to be asking is a direct question: I'm wondering, "What is the time?"

This site gives instructions on how to make an indirect question. And this one an explanation of how embedded questions are formed.

Perhaps you could use ellipsis.

In reported speech, the ellipsis is sometimes used to represent an intentional silence, perhaps indicating irritation, dismay, shock or disgust. This definition is more known with younger, internet savvy generations.

or with a different use from the same Wiki page:

The ellipsis is one of the favorite constructions of Internet chat rooms, and it has evolved over the past ten years into a staple of text-messaging. Although an ellipsis is technically complete with three periods (...), its rise in popularity as a "trailing-off" or "silence" indicator, particularly in mid-20th century comic strip and comic book prose writing, has led to expanded uses online. It has been used in new ways online, sometimes at the end of a message "to signal that the rest of the message is forthcoming.

I understand your dilemma, but putting is before the benefit is not a good idea. In an indirect question, there is no subject-verb inversion:

• Normal: you were here (subject, finite verb).

• Direct question: she asked: "where were you?" (finite verb, subject).

• Indirect question: she asked where you were (subject, finite verb).

In speech, you can't hear punctuation, and so only the tone at the end of the sentence (it goes up only in direct questions) and the word order differ.

There are also indirect questions embedded within direct questions:

• Did you ask her | where she was?

Here the tone goes up because the main clause is a direct question (hence the question mark); and so word order is the only difference between the two. As a result, some people are a bit sloppy with word order in indirect questions in writing and use inversion; but this is not recommended by style books, nor by most educated speakers. But you will see it occasionally, especially in longer sentences.

In your example I agree that putting is at the end makes the sentence unwieldy. The solution would be to move is more to the front, but not before the subject:

I don't know what the benefit is of closing schools when there's a flu outbreak.

Or:

I don't know what the benefit of closing schools is when there's a flu outbreak.