The transitive verb form “to have got” has the same meaning as “to have” – to be in possession of/hold/own. In it, have is an auxiliary verb, but not a modal auxiliary. A model always takes an infinitive.
I’m very busy. I have / I’ve got a lot on my plate at the moment.
He loves motorbikes. He has / He’s got three of them.
But the intransitive verb form “to have got to” has the same meaning as “to haveto”, and both of these work in the same way as the modal verb must. Therefore, like must, in this sense both have to and have got to always take a bare infinitive.
We must leave immediately! = We have to leave immediately! = We’ve got to leave immediately!
He can’t come out with you tonight. He has to / has got to be somewhere.