Now this has nothing to do with the Unfortunate Lad whatsoever, but one thing I noticed is that the chorus of "St James' Infirmary Blues" also exists as a separate song.

Here's an example of this chorus, from Carl Sandburg's American Songbag "Those Gambler Blues":

Let her go, let her go, God bless her,
Wherever she may be;
There'll never be another like her,
There'll never be another for me.

Here's an example of a song with that floating verse:
If He's Gone Let Him Go, God Bless Him - Ollie Gilbert, Mountain View, Arkansas on May 26, 1969

Verse:
If he's gone let 'im go, God bless 'im
Wherever he may be
He may roam this wide world over
He'll find no other like me

No Girl Like Me - Holly Hodges, Prairie Grove, Ark. Dec. 17, 1960

Verse:
Since he's gone, let him go, God bless him
He is mine wherever he may be
He may ramble Arkansas over
And he'll find no girl like me.

Here's another, from Doug Wallin's "Let her Go, Let her Go" on the album "Far in the Mountains - Volume 5 (MTCD513)":
Let her go, let her go, God bless her
She’s nothing no more to me
For God in Heaven knows, love
It’ll be alright some day

The verse may have originated with the song "Fare You Well Cold Winter", also known as "Farewell He", here are some examples:
Adieu to Cold Winter - Mr. Frank Pool, Fayetteville, Arkansas on January 6, 1958

Verses:
If he's gone, let him go
Let him sink or let him swim
If he don't care for me
I'm sure I don't for him
And I wish himself a fortune
And myself, a better grace
And I'll catch another
In a fair and closer place

Reba Jenkins, Wheatland, Missouri on January 27, 1973

Verse:

My love is on the ocean
He can sink or he can swim
He don't care for me
An' I'm sure I don't for him
There's plenty more without him
As nice young me as he
An' I can find another
Since he's gone back on me

Let Him Go, Let Him Tarry - Tom Lenihan, Knockbrack, Miltown Malbay, Ireland, March 1988

Verse:
Let him go, let him tarry, let him sink or let him swim.
He doesn't care for me, nor I don't care for him.
He can go and get another, I hope he will enjoy,
For I'm going to marry a far nicer boy.

There's yet another song from the British Isles, "Go and Leave Me / Fond Affection / Dear Companion" which has a similar verse:

From the Carter Family's version of "Dear Companion":
Just go and leave me if you wish to
It will never trouble me
For in your heart you love another
And in my grave I'd rather be

From Percy Webb's "Go and Leae Me" on the album "King's Head Folk Club (MTCD356-7)":
So go and leave me if you wish, love
Never let me cross your mind
For if you think I'm so unworthy
Go and leave me I don't mind

I know this is probably meaningless, it's just a floating verse, after all, but I wanted to point it out.