The class was composed of thirty students, including Jonathan and Kelly.
In this sentence, the prepositional phrase ‘including Jonathan and Kelly’ is a non-restrictive element in the clause structure (a supplementary adjunct). It does not modify ‘thirty students’ but provides additional information about them.
Can we call this a predicative adjunct? I am familiar with the concept of ‘supplementary adjectives’, which are said to be classified this way (see this forum discussion). They, too, provide information about a noun phrase.
The class was composed of thirty students, including Jonathan and
No, it’s not a predicative adjunct since it doesn’t relate to a predicand. Compare, for example, Unwilling to accept these terms, Max resigned, where the AdjP "unwilling to accept these terms" is an adjunct in clause structure and it is also predicative in that it relates to a predicand, i.e. "Max".
In your example, the expression including Jonathan and Kelly does not relate to a predicand. It does not provide ascriptive information about "the class" or "thirty students".
The received wisdom seems to allow for two possible interpretations:
"Including" is a preposition, so "including Jonathan and Kelly" is simply a preposition phrase functioning as an adjunct, probably a supplementary one.
"Including" is reanalysed as a marginal member of the coordinator category, so it would be an NP functioning as second coordinate in the NP coordination "thirty students, including Jonathan and Kelly". This interpretation is weakened by the fact that it can be fronted, as in Including Jonathan and Kelly, the class was composed of thirty students.