No. The negative of 'have got to ' is 'haven't got to'. This is because the verb 'have' is an auxiliary verb here. For example:

"Have we got to go to school tomorrow?"

"No, we haven't got to go to school tomorrow."

This is less common than the standard "Do we have to..?" and "We don't have to..".  And as you pointed out, the "got" variation is also more informal.

"Got" isn't specifically British, though. To express obligation, informal American English uses "got to" just as much as British English does.

no you can 't say that....

it has to be "didn't have to"

e.g. I didn't have to go to work today, so I stayed at home

- Lawrence (uk)

@Su.Ki. 

I never know if I should take the form "Tempy" as a term of endearment. :)  I'm having trouble coming up with one for you.  "SuKiOoKi?"

"I've got to go" is fine as an informal way to say "I have to go."  The other two rub me the wrong way.  I know actual people would use them, but I am not one of them.  To each his own.

Tempy, sorry....maybe my comment wasn't clear. I'm aware that "haven't got to go"  is what you might call a "Britishism". 

What I meant in my final sentence was that the use of "got" to indicate obligation, necessity or urgency is common in American English. With increasing levels of informality:

I've got to go -->

I got to go -->

I gotta go.

Isn't that the case?