"I have got" and "I haven't got" are equivalent to simple present in meaning (= "I have" and "I haven't" respectively). But in terms of form or structure, they are clearly present perfect. "Got" here is formally a past participle (also called a perfect participle or passive participle).
Rules on backshift of tenses in reported speech
In reported speech, a simple present becomes a simple past, and a present perfect becomes a past perfect.
So the present perfect "I have got" becomes the past perfect "I had got". "I haven't got" becomes "I hadn't got" in reported speech.
Of course, instead of remembering two rules (that the simple present becomes the simple past, and that the present perfect becomes the past perfect in reported speech), you could ignore the participles completely and just remember that the present becomes the past in reported speech.
Reported speech is allowed to paraphrase. If someone says "I haven't got any money", I can choose whether to report that as "he said he hadn't got any money", "he said he hadn't any money" or "he said he didn't have any money", among other alternatives. In fact, if someone says "I ain't got no dough", it's still possible for me to report that as "he said he hadn't got any money".
The Cambridge Grammar of the English Language (Huddleston and Pullum) confirms that "had got" is less commonly found than "have got", but this isn't the same thing as saying it's never used or is unacceptable. I find it perfectly acceptable in the sentence "he said he hadn't got any money", and I can imagine I might utter such a sentence myself.
CGEL implies that you are right to regard to "have got" as equivalent to a simple present. While it notes that "have got" derives from the perfect, it also describes "had got" as the preterite form. "Preterite" is an exact synonym for what you call "past simple". So there you have it. As I've said before, you must either regard both "have got" and "had got" as simple, or you must regard both as perfect. You can't say that "have got" is simple but "had got" perfect; that denies the connection between them.
(Of course, in certain sentences, where the meaning is "have obtained"/"had obtained", they are semantically perfect as well as formally perfect. In that usage, AmE uses "gotten".)
Rarity of "had got" form - with crucial exceptions
You mentioned that your workbooks state that "have got" must become "had" in the past. This (at least the compulsory nature of it) is explicitly denied by CGEL. However, the workbooks' view receives some support from A Practical English Grammar (Oxford University Press, 1986). It lists "have (got)" as the present, but only "had" as the past. However, here is a crucial point: it lists the negative "haven't (got)" as having a corresponding past form "hadn't (got)", and the interrogative "have you (etc) (got)" as having a corresponding past form "had you (etc) (got)". So it explicitly recognises the possibility of including the "got" after negative and interrogative "had"s, just not after affirmative ones.
In "hadn't got any money", of course, "haven't got" and "hadn't got" are negative and therefore are allowed by the Practical English Grammar.