Is it acceptable to use 'that' followed by a direct quote?

New York Times:

Even before President Obama declared this month that “I have Israel’s back” in its escalating confrontation with Iran, pro-Israel figures like the evangelical Christian leader Gary L. Bauer and the conservative commentator William Kristol were pushing for more.

Is this sentence, in particular the part in bold, grammatical? I imagine that it would be impossible for such constructs not to occasionally slip in spoken conversations, but are they formal/suitable for papers?

To be redundantly specific, I'm asking whether or not the following would be more natural:

... President Obama declared this month that “He had Israel’s back”...

Or is the original sentence perfectly normal and formal?

As @Robusto implies, the usage is perfectly grammatical. Few would quibble with...

The president declared that “the first to benefit from a free Iraq would be the Iraqi people, themselves"

...followed on the very same page by two more instances of the same construction, in Terry H. Anderson's 2011 Bush's wars (here's the full transcript, showing those words are verbatim).

I and many others find it "clumsy" in the specific case where the quoted text starts with "I", because in a spoken context we can't see the quote marks, so we may misinterpret "I" as referring to the speaker/reporter. Just because it's technically "valid", doesn't mean it's good writing style.

Having said that, here's another link to two instances of declared that "I..." on a single page. I may not much like the usage, but obviously some people don't have a problem with it.

OP has actually asked three specific questions, so here are the specific answers...

• is it "acceptable"? - Yes, to some - but not to all, particularly in a spoken context

• is it "grammatical"? - Yes

• is OP's rephrasing more "natural"? - Yes ("-ish" - I agree absolutely ruakh's comment)