I presume that you have gone onto the wrong track. There literally exist 2 verb phrases here. What I mean is that one is given 'inflection' (ie. Subject Agreement + Tense + Aspect) and the other only given 'Tense' (and all that goes with it: Tense, Agreement with Object, etc.)
To make it down to earth, the phrase (actually the entire Agreement Phrase, SPEC exclusive), disregarding peculiarities in tense and aspect assignment, resembles: have to eat. What appears 'larger' is not the semantic features of the Verb Phrase, but actually the functional features of it. I would illustrate the 'mega phrase' this way (Move and Merge are ignored, for simplicity's sake. Only spelt out product is considered):
[ Agr P [SPEC omitted] [Agr' [Agr, Mood: conditional, Aspect: perfect] [VP would have had] [TP [T to, Aspect: Perfective, Progressive] [VP have been eating]]
Where the Main Verb 'have' (represented as 'had' in the VP shell) takes a TP as its obligatory complement.
I don't think 'have to' is a modal verb, but closer to a Main Verb taking TP as an obligatory complement, because in modal verbs, tense is usually covert (save 'can').