We're having the proper chant before the opening hymn, ostensibly with all verses included. We're using Pitoni's setting as the Offertory motet.

I may be mistaken, but Adam's assignment of psalm verses to the SEP antiphons aren't necessarily the only verse options that could alternate.
That said, I think the congregation would be all the better for hearing both the English translation with verses and then the polyphonic "off spring."
Other forms of this consistent "stuffing" would have a congregation joining in "Shepherd Me, O God" after the choir sings "Ego sum pastor bonis" as oppossed to the Communio followed by some random text/tune that emphasizes the act of receiving Communion. YMMV.

If one wished to do a polyphonic motet and felt that any missing ps. vv. were needed, he could always do the vv. to a ps. tone that matched the motet, and then repeat the motet. This, I think, would be a rather historical pattern: i.e., we know that some of Tallis' motets require chanted vv., and his are no exception.
On the other hand, we also know that in times past 'they' were not timid about omitting text that 'they' thought took too much time. (Which only affirms that 'their' example is not always one to be followed.)

There's nothing wrong with what you describe, YR: a prelude and then an entrance hymn/ song, option 4 in the GIRM. The polyphonic prelude with most of the Introit text is a hat-tip to tradition. And you're right- it's a joyful motet for the season.
I won't comment on the particular hymn/ song.

Palestrina, Byrd and other greats of the Catholic tradition did not fuss over every verse of every proper, and neither should we.

The only question about licitude would involve the text, and that's only if your local practice is to sing the Proper text, the whole Proper text, and only the Proper text. A motet with a not-quite-accurate/complete Proper text would probably rate higher than "cantus alia apta" but it would at the least be "cantus alia apta". I see the given options for the Propers to be a wonderful gift of freedom. Now, freedom is not license to sing any old hymn. But OTOH, having 2 different words in a Proper setting is not going to invalidate the Consecration. It's the liturgy; it's not a spell in ceremonial magic where the demons will cross the Triangle of Art and eat your soul if you miss a word.

Ummm, Charles, did you misread JQ? He wrote (emphasis mine):

    Now, freedom is not license to sing any old hymn.

Me?! I'm more confused than ever. I thought Croce wrote Time in a Bottle. What is this Latin gibberish?

Geez, Chuck, I just bought those appelations from a so-called wine boutique in Fayetteville, NC last summer, notably the PN and a Malbec. I do my homework. The "shoppe" was situated next to a "Dollar Store." My cousin steered me there, I shoulda known bettah.
I'm not a cheap date, Chuck, just ask JT, AOZ, Kathy or even Mahrt. I got Yoda facetime two days straight with Bill by offering him a 2000 Mondavi Reserve Cab the first night in Loyola, and a boutique Cab (Regucsci) from Napa the second night in the dorms.
It's the only reason CMAA folk tolerate me at all, is my California (neo-Francaise) nose. Extremely red.

Charles, you're not popular because of the wine. Heck, wine gives me migraines. You're the real deal, a musician and a church working man, and you're a good guy. Go to Colloquium without bottles once (say, isn't there a Colloquium coming up in a month or two?) and see what happens. Even if you don't have a two-piece custom-made pool cue.

YR, been there, understand that. :)

Charles, you are lovable with or without the wine, as Wendy and all your CMAA women agree. Now if I could only take the whine out of some of my sons...

I'm doing the Pitoni as well, but as the Anglican calendar is slightly different, I'm using it next week.

I'd like to associate myself with Jeff Q's comments above. Just do it in place of the Introit and let that be that (pastoral considerations, of course, being observed).

ScottKChicago, was that legendary tunester at Resurrection Parish able to redeem bad, bad Leroy Brown through his strummings, as well? Perhaps he was able to craft a kinder, gentler Leroy in his lyrics?

I must admit that singing an introit first and then going into a processional hymn seems to me to be "putting the cart before the horse." There are those celebrations when I think in certain parishes an appropriate processional hymn can be justified. Then as the procession nears the altar, go to the proper introit which reflects the spirit & theme of the feast's proper. This can be very effective especially with a lengthy procession and a celebrant who incenses with "noble solemnity." I try, when possible, to select motets based on the text of the proper offertorium, which the choir sings after the chant. We have a great treasure from Palestrina of the Offertoria for the liturgical year. We are learning them as we are able and can balance the parts. Also, when possible, after the communio with psalm verses (Thanks, Mr. Rice, for doing the work for us!) a motet setting either of the communio text or of verses of the suggested psalm- there are
some small gems from Isaac and di Lasso, etc. O, that we might be able to do it consistently!

Charles, you are so wise. We must teach the children and then let them teach the adults. I finally got to listen to the wonderful Choir & Schola from Saint Stephen in Sac. Those children, like the children from the Madeleine in SLC, are our hope for the future! Those fortunate enough to go to the Colloquiam this year are in for a real treat. My wife and I heard them at the Cathedral of Christ the Light earlier this spring. Our next treat was recently hearing the Choir of Canterbury Cathedral at Grace Cathedral in SFO. O that we would really invest in education. Keep up the good work in Visalia!