I assume (and hope) that you want the mode I tune associated with Crux fidelis.

In the event that this isn't essential to you, you may find 'Sing, my tongue...' with Crux fidelis as the fourth stanza in round note heads to the familiar mode III Sarum Pange lingua melody at no. 166 of The Hymnal 1982. The translation is the same as in The Anglican Use Gradual.

In case anyone wants this, here is the Picardy/Crux Fidelis amalgam in modern notation, with the AUG translation.

To go off at a tangent: there is a rather fine hymn-meditation by Kevin Nichols on 'Crux fidelis'. Mgr Nichols (d.2006) lived in retirement at our church and published several poems and hymns there. Feel free to use it.

1. Faithful Cross, sustain your burden,
Do not splinter, do not crack.
Though the load of all our sorrows
Hangs a dead weight on your back;
Upright on a hill of sadness
In the gale of evil’s power.
Hold him strongly, hold him gently
At his covenanted hour.

2. Nameless in the forest mounting,
Shoot and sapling, branch and tree,
Felled, dismembered, planed and jointed
For this day’s dark mystery.
Gibbet, infamous, ennobled
By this death and by this birth,
Hold your cross-grained branches open,
Harbour for a shipwrecked earth.

3. When, the noontide darkness ending,
He whom you have borne is dead,
In his mother’s arms laid gently,
You are left untenanted:
Sharp against the soul’s horizon
Still uphold us, shining tree,
Emblem of the Saviour’s passion,
Standard of his victory.

—words: ‘Crux Fidelis’ by Kevin Nichols (1929–2006),
priest-poet of St Mary’s Cathedral, Newcastle upon Tyne,
from Decryptions published at the Cathedral in 2006;
music: traditional French melody,
adapted by Giovanni Pergolesi (1710–1836)

I used the melody found in the AUG, arranged by C. David Burt, copyright 2004. Lines 3 and 5 presented an interesting conundrum: the relentlessly trochaic meter of this translation seemed to me finicky when applied to the standard melody--more so in line 3. For this reason I fudged the beat pattern in line 3, though I remain unconvinced that this was the best solution, as I think Mr. Osborn rightly discerns.

In general I tried to point the rest of the melody faithfully to the chant, though in a way which was standard across all 10 verses, in an effort to increase the accessibility of the melody for my choir.

Are there more than one version of the AUG? The online one seems to have a couple textual variants from your score. Just curious. Beautiful, by the way! Thank you, it saves me a lot of work.