It's possible, Torquemada, that the instrumental parts were produced by Chanticleer's arranger and are unpublished.

FWIW, Fr. Krisman, CPDL categorizes it among madrigals.

What you hear on the recording is the vocal parts doubled by instruments, (violins in the upper parts), with the bass line treated as continuo with theorbo [EDIT--it's actually a harp] and (probably) viola da gamba (which goes rogue for a few cool little ornaments in places.) The introduction is an excerpted phrase which serves rather well in its place. Given the information you have (which is likely all the printed information there is), you could easily work out your own roadmap with the instruments you have and the combinations you like.

(May I use this opportunity to urge VERY STRONGLY that the organists among us take time to learn at least the basics of continuo playing, not as some sort of archaic novelty but as a practical skill? Much excellent repertoire will be open to you, and you won't be tied to one editor's plus-or-minus idea of a realization.)

Also, I suspect the genre we're all looking for is villancico.

I think there is a lot more going on in the Chanticler version than added instruments; the CPDL version seems to be a by-ear transcription although, since CPDL doesn't insist on sources, one is often taking someone's word for it that what is offered is actually in public domain.

A google search turns up a detailed footnote in Russel's From Serra to Sancho: Music and Pageantry in the California Missions and even more usefully (figure 5) a facsimile.

Thank you all so, so much for your kind assistance and resources! They are immensely appreciated!