It's not a hymn, but Byrd's Sacerdotes Domini would be splendid as a motet at the offertory.

I expect this text is under copyright, but I have no idea who the author is. I've heard it sung to Old Hundredth. It's probably also meant for the Chrism Mass

EDIT: found more info through some better searching

(Organists may love ABBOT'S LEIGH but the tune as such is better thought of as a vocalise exercise for Catholic congregations.)

There is a hymn "Our Pastor" that appears in one of the old Berge's Catholic hymnals.

" . . . it’s definitely possible that the sort of Catholic who attends/seeks out these masses is more likely than average to be willing/able to sing at mass in the first place."

Gold star answer. Because 8.7.8.7 D tunes are long by nature, there's a built-in problem of developing pleasing variation over the duration of a verse/stanza; but there's something of a hidden (that is to the skilled musician) danger for unskilled PIPs (particularly singing earlier in the morning) in merciless voice leading and tessitura*. Many 8.7.8.7 D (or LMD, CMD) tunes may risk this. Certain masters found a kind of balance: consider Haydn's AUSTRIA.

* Re tessitura: HELMSLEY is an oddball meter, perhaps thought of as 8.7.8.7.12.7 instead of 8.7.8.7.[4.4.4.]7, but what makes it just a bit easier is that it's *not* a Double meter and the compass of the melody of the 4.4.4. line is fairly narrow within a comfortable part of the compass of unskilled mezzo/baritone voices (who are the bulk of singers in a congregation) and with excellent voice leading, so that the unskilled singer gets some relaxation before re-entry for the next verse/stanza.

Another Hymn

Words by Robert Loretz
Music by Jean Sibelius
Tune: Be Still my Soul

1. Eternal Priest, You have become the Temple
Entered the tent where God longs to meet man,
and there you offer through your flesh surrendered
your very self, as Sacrificial Lamb,
And in your hour, at last embrace as Husband
Her, ever loved in your eternal plan

2. Now intercede, fill with your Holy Spirit
him whom you raise to share the rank of priest,
That by his mouth your gospel may be spoken
And from his hand your flock may know your peace,
That he may worthily approach your altar
Off'ring through you the Eucharistic feast

3. Conform his heart to yours, Eternal Shepherd
That he may lay his life down for the lost
May he partake, O Christ, in your own Headship
And feed your Body, counting not the cost
Configure him to have Your Heart, O Husband
Which for your Bride was opened on the cross

4. Glory to Christ, our Altar, Priest, and Victim,
In whose self-gift His Bride has been restored
Praise to the the Spirit given through His dying
In whom she is one spirit with the Lord,
So that the Father for whom both are given
Be, in the Church, eternally adored.