Aha. This question is very interesting and unless one has thought about it or taught English, the answer is not obvious.
Here we go: In English (all varieties), there are two ways to say have in the sense of possess in the simple present tense:
I have a car and I have got a car.
Despite what people may try to say via google et alia both forms in the present simple are used in BrE and AmE though one might be tempted to say differently.
(Some time ago, some lady who thought she could teach grammar on a very well-known professional site on the Internet was claiming that phrases like "Got Milk" used in a now famous ad were bad grammar. In fact, "Got Milk" is just a shortening of: [Have] you got milk? In speech, the have is sometimes dropped.
Here's the RUB: The simple past tense in BOTH is: had.
For example: I had a car last week. I didn't have a car last week. Did I have a car last week? The auxiliary is did.
BUT: The present perfect tense is not the same (repeat, not the same) in BrE and AmE. For example:
He's got a lot of money recently. [verb: to own or possess] [BrE]
He's gotten a lot of money recently. [verb: to own or possess] [AmE].
In the present perfect tense, Brits use have got regardless of the fact they use either have got and have in the present and Americans use have gotten. The participle gotten is not used in modern BrE.
Therefore, the British use of have got can be present simple OR present perfect.
When you first hear this, unless you realize what is going on, you might mistake the tenses here if you have not been told that gotten is not used in BrE.
To an American (trained and/or educated ear), when you hear the BrE speakers use have got as a present perfect tense, you feel as if something is missing. It is not. That's just the way it is.
All the tenses EXCEPT for the present prefect with gotten in AmE are exactly the same on both sides of the pond.
Modals; generally, yes, the modals used with the verb have meaning to own or possess) in the present are used like this:
He must/can/might/should/could/may have [possess] some milk in the fridge. For own or possess, only HAVE, no have got.
But careful, with the verb GET: must have got, should have got, could have got [past tense] in BrE would be: must have gotten, should have gotten, etc. in AmE.
So when get is used in the present perfect as BUY (or other meanings of get) something, for example, one can have:
He must have got [bought] some wine for the party. [BrE]
He must have gotten [bought] some wine for the party. [AmE]
This is the verb get, but that can be confusing if one is not aware of it. Don't confuse the present perfect of GET [have got] with the present tense have/have got (possess).
As for the infinitive: I have got to buy a car or I have to buy a car are both the same meaning. Both are present simple and both are AmE or BrE. But the meaning here is DIFFERENT. To have to [do something]; have got to [do something]. That is, be obliged to do something.