Strange thing is that I can't come up with other similar examples for 'ik heb het...' At least you could say in the case of koud/ warm/ heet/ benauwd, that it doesn't say something about the person himself, but about the feeling he has. Has. But, to make it all more confusing, with every other feeling I can think of, you use "zijn". BTW you could use "zijn" for koud/ warm/ heet, but then you are talking about the literal body temperature or about your attitude towards other people...
I know in English it just sounds crazy. I'm an Spanish speaker and I can tell you in Spanish we also use "have" for those things, and then this is familiar to us, and it happens the same with German, French, Italian, Portuguese, as far as I know. I interprete it this way: warm is not working as an adjective (like HOT) but as a noun (HEAT), so in somehow this means you "have" some HEAT in your body.
You can say "Ik heb het warm" using normal word order (Subject+working verb+...+infinitive/participle) but in this case, i believe that because of "At the moment" or "momenteel" It falls into a different word order category. In that order it starts with First element of sentence+ conjugated verb + subject+...+infinitive/participle.
There is a great bit in the Discworld novel "Guards, Guards":
"I shall deal with the matter momentarily," the Patrician said. It was a good word. It always made people hesitate. They were never quite sure whether he meant he'd deal with it now, or just deal with it briefly.
I think momentarily is too vague to serve here.