Note: Not a direct answer, but...
According to A.G. Rigg, 'Morphologoy and Syntax' in Mantello and Rigg, eds, Medieval Latin: An Introduction and Bibliographical Guide (Washington, DC, 1996), 85:
In Classical Latin the past participle is sometimes (though rarely) used predicatively after habere: domitas habere libidines, "to have one's desires tamed," i.e. "to have tamed one's desires." From this it is an easy step to the French Je l'ai tué, "I have killed him." English developed I have killed him in the same way, not from French influence but from the senses inherent in have and the past participle. When Medieval Latin uses such constructions (habere plus perfect participle to form a
transitive perfect tense), it is probably in imitation of the vernacular rather than of the rare Classical Latin construction.
However, I'd say the more common medieval 'analytic' forms involved a shift in the way the passive voice might be written:
present scribitur scriptus, -a, -um est
perfect scriptus, -a, -um est scriptus, -a, -um fuit
pluperfect scriptus, -a, -um erat scriptus, -a, -um fuerat
future perfect scriptus, -a, -um erit scriptus, -a, -um fuerit
present scribatur scriptus, -a, -um sit
perfect scriptus, -a, -um sit scriptus, -a, -um fuerit
pluperfect scriptus, -a, -um erat scriptus, -a, -um fuisset
Compare Rigg, ibid.:
In Classical Latin the perfect passive is formed by esse and the past participle (iussus est, "he was ordered"); in Medieval Latin the verb esse sometimes regains its literal tense, so that amata est can mean "she is loved"; consequently, to form the past, past tenses of esse are needed (amata erat/fuit); this is a natural consequence of the adjectival nature of the past participle.
When reading post-classical texts, then, one must be careful about this shift. In particular, one must watch for ambiguity in three cases:
Classical perfect indicative vs ‘medieval’ present indicative →
scriptus, -a, -um est;
Classical perfect subjunctive vs ‘medieval’ present subjuctive →
scriptus, -a, -um sit; and
(as is regularly the case) ‘medieval’ future perfect indicative and
‘medieval’ perfect subjunctive →
scriptus, -a, -um fuerit.
And, certainly, a 'medieval' author might use both the medieval and classical form in one and the same work. Whether that is because there is some special distinction that can be drawn between the two uses is hard to say. But I doubt that there was a universal, but unstated, distinction that all authors drew upon. (Indeed, I'd be inclined to wonder if the different uses might not be attributable to what we might call palaeographical reasons.)
Finally, another avenue of inquiry might be the adoption of the Greek periphrastic in, e.g., the Vulgate:
Et ecce: eris tacens et non poteris loqui usque in diem, quo haec fiant... (Lc 1:20)
The Latin construction of the Greek periphrastic is a present participle + some tense of esse (present, imperfect, or future). But I don't know, off-hand, what connection the above analytic forms have to the adoption of this type of periphrastic construction....