I think it may not be an accident that the earliest reference you found to Encaenia composition comes from the late 17th century. From what I've been able to find, music was not a major part of university ceremonial--in fact, many universities were actively hostile to music in general.*
From Life in the Medieval University:
The principle on which modern Deans of colleges have sometimes decided
that "gramophones are dogs" and therefore to be excluded from College,
can be traced in numerous regulations against musical instruments,
which disturb the peace essential to learning.
If scholars are found bearing arms by day in the students' quarter of
the town, they are to forfeit their arms, and if they are found at
night with either arms or musical instruments in the students'
quarter, they are to forfeit arms or instruments. If they are found
outside their own quarters, by night or by day, with arms or musical
instruments, the town officials will deal with laymen, and the Bishop
or the Rector with clerks.
Instead of the whole-class convocations of today, the ceremonies of Inception and Determination were not lively affairs. Students would graduate in small groups as they were deemed ready. According to this history of medieval universities, these ceremonies involved the faculty grilling would-be graduates on texts, a "peculiarly solemn disputation known as 'Vespers'" (452), and then the inceptor's formal inaugural lecture.
The sole reference to celebratory music in the description of the University of Paris' Inception Ceremony is this:
The evening concluded with a banquet given at the expense of the
Inceptor or a party of Inceptors to the Masters and others, at which
it is probable that the prohibitions which we find in some
Universities against dancing or the introduction of actors and
trumpeters were not always strictly complied with (453).
In short, it seems unlikely to me that music would have been a part of formal university ceremonial in this era. Attitudes toward music would need to loosen, and graduation ceremonies needed time to evolve into celebratory rituals. However, note that the sources I found were mostly concerned with English and French universities. It is of course possible that, say, Spanish or Italian universities had different customs.
* Music may have been part of the quadrivium, but the sources cited say that it was given scant attention. Consistent with university-wide bans on instruments, instruction in music had more to do with theory, logic, and even mysticism than it did with performance.