You generally have two types of "have":
(1) He has two sons. (stative)
(2) He has lunch alone. (dynamic)
A stative "have" can be followed by "got", whereas a dynamic "have" cannot:
(1a) He has got two sons. (stative)
(2a) *He has got lunch alone. (dynamic)
A dynamic "have" can appear in the progressive, whereas a stative "have" cannot:
(1b) *He is having two sons. (stative)
(2b) He is having lunch alone. (dynamic)
Now, is the following "have" stative or dynamic?
(3) I have to go.
(3a) I have got to go.
(3b) I'm having to go.
Best Answer
It seems that -- if I'm understanding CGEL right -- that your examples are all using the stative "HAVE" (not the dynamic "HAVE"). And that it is a lexical verb in your 1st and 3rd versions (#3 and #3b), while it is an auxiliary verb in your 2nd version (#3a) due to the "have got".
I intend to show some excerpts from CGEL that is directly related to this topic, but first I'll have to provide some background info (that's also in CGEL). For instance, some background info on the properties of auxiliary verbs and modal auxiliary verbs.
Auxiliary verbs have syntactic properties that distinguish them from the open class of lexical verbs; and modal auxiliaries have an additional set of properties that distinguish them from the other auxiliaries (CGEL page 92). A quick summary and examples of these properties follow, which use "WILL" for the examples. CGEL page 108:
In general, the auxiliary verbs will have the properties [A-H], and the modal auxiliaries will have those properties and the additional properties [I-M]. Although the prototypical central modal auxiliary verbs will have the properties [A-M], there are some modal auxiliaries that don't have all of them, e.g. "must".
Let's first see some info about the dynamic "HAVE". The dynamic "HAVE" is a lexical verb (not an auxiliary). CGEL page 111:
The constructions that involve "have to V" use the stative "HAVE"; and those usages are of a lexical verb and not an auxiliary (except for some dialects). CGEL page 111-2:
That above info about [57.iii] seems to be directly related to what you were asking about in your post (because it uses the "have to V" construction). Also, note the progressive example: "I'm having to work late tonight" (which has a form similar to your 3rd example #3b).
Within this same section of the CGEL, there is this following bit which might be helpful too. Page 112:
This following excerpt also has info related to the OP's topic. On page 113:
And then there's the topic of modality, where "HAVE" is involved too, as a lexical modal (and as an auxiliary in the idiom "have got to V"); and there's also a "dynamic" term involved: deontic vs dynamic vs epistemic. CGEL page 205-6:
In that above excerpt, those uses of "HAVE" are of the stative "HAVE" (especially since they have alternations with "have got"), and they all involve the construction(s) "have (got) to V".
So in conclusion, if I'm understanding all this stuff in this post properly, it seems that the OP's three examples (#3, #3a, #3b) are all using the stative "HAVE"; and that those versions that use "HAVE got", the "HAVE" is an auxiliary verb, otherwise it is a lexical verb (and also a lexical modal). And all three involve strong modality due to using either the lexical modal "HAVE" or the idiom "have got" in their "have (got) to V" constructions.
NOTE: CGEL is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language (CGEL).