I would have liked to use the introit chant myself, but the selection is a legitimate application of "et alius cantus aptus." It was a bit tricky because we've got everything from very Traditional Catholics, right through to evangelical protestants attending. The couple are both college friends of mine. I organised the confirmation mass for one of them, who is a very recent convert to Catholicism. I organised their confirmation mass in the college chapel last year, and we stuck with principally English for the music as their parents were strongly anti-catholic.

According to the book by Bishop Peter Elliot on the Ceremonies of the Modern Roman Rite, there are several customary ways to begin the nuptial:

1.) Celebrant greets the bride and the groom at the door and they enter with the sacred ministers.

2.) Sacred ministers enter in procession to either silence or soft instrumental music and incense the altar. The bride enters with the introit chant and meets the groom at or near the sanctuary.

There is some other permission to allow for local customs. A number of weddings I have attended here in Sydney have had the introit chanted for the entrance of the sacred ministers and then some other music for the entrance of the bride, so it seems to be something of a local custom.

I'm somewhat inclined to agree with having the introit for the entrance of the sacred ministers and then a separate instrumental pieces for the entrance of the bride, although it does not strictly adhere to the books.

The couple agreed that we should sing the Gregorian Chant with it's attendant Psalm verses until about half the people have received communion (or a blessing) then move into the thanksgiving hymn.

There's a fair bit riding on this as it is the first wedding I have done as a "Director of Music."

It's on a Friday during Ordinary Time, and we do have a strict time limit on the use of the church.

ICEL Gloria, with festive accompaniment (at a good clip), perhaps? A few seconds longer than saying it.
Please think about it, given the importance of the Gloria.

There hangs a question as to whether or not the Kyrie is actually a part of the penitential act. It has been the source of much debate and discussion.

The fact that people debate and discuss something has little to do with whether a matter is plain as all get out, which this issue is, if only people would stop being so ridiculous.

The Kyrie CAN be a part of the Penitential Act, and it can also not be a part of the Penitential Act. When it is not, it is said (or sung) separately.

the lack of a Gloria is mainly on the advice of a leading church musician and liturgist

Does this person not (how to put this delicately?) read the Missal?

Well, the person in question is from the office of liturgy. It may be that this hasn't come up with the New Edition of the Missal. If we have to insert it, we can always just chant the Missa de Angelis Gloria.

The Oxford Book of Wedding Music for Manuals has a good Canon in D

Isn't the Pachelbel in the Diane Bish wedding book? I don't have my copy near the computer but I thought it was in there.

For some off the beaten path pieces:

Jonathan Dove: My love is mine

https://www.youtube.com/watch?v=ONjbEyn5qMM

Monteverdi: Ego flos campi

https://www.youtube.com/watch?v=Q-3WP1ZXCi8

Both have texts from the Song of Solomon so no difficulty there.

Thanks for this, Francis.

https://imslp.org/wiki/A_Wedding_Processional_(Sowerby%2C_Leo)