The tune to which the DT version is sung by Lillis Ó Laoire is indeed a variation of The Ten Penny Bit. It is easy to find sheet music for this jig; I have it in Allan's "Irish Fiddler" arranged by Hugh McDermott and published in Glasgow by Mozart Allan. The dance tune is not exactly as I would sing it, particularly the second half, but it is clearly recognisable as a version of the tune.
The three sets of lyrics for Cearc agus Coileach seem to all fit all the tunes given.
Lillis got the song An Chearc is an Coileach from John Ghráinne Ó Duibheannaigh (Rann na Feirste) & Antain Mag Shamhráin, and recorded it on Bláth Gach Géag dá dTig, Cló Iar-Chonnachta, 1992.

Donal O'Sullivan published a longer Munster version in a section on "humorous songs". Even if it's sung in jig-time, I don't think the hen's lamenting the death of her mate is funny just because they are chickens! Also, it could be a political allegory, like the Blackbird of Sweet Avondale (Parnell) and Nell Flaherty's Drake (Emmet, and a song that sounds humourous in a way). I especially wonder about that possibility given the lines about spending time in gaol, which otherwise don't seem relevant to the rest of the song.

Donal O'Sullivan, Songs of the Irish. Cork: Mercier,1981 (1rst edition 1960)

CEARC AGUS COILEACH

O! Cearc agus coileach a d'imigh le chéile
'S do shiúladar Éire gur briseadh a gcroí
Do chaitheadar sealad i Sligeach i ngeibheann
Gur chuireadar sgéala i gcoinne lucht dlí.
Do bhí siad annsin i lár barra 'gus binns,
'S duine gan chéill a chur ionnta sgaol,
Gur ag Liam Mhac an Ailigh atá siad ar fearach
'G ithe mónóga sléibhe 's 'na gcodla sa' bhfraoch

[cearc]
Da bhfeicfeá mo choileach lá aonach na Sráide
'S a ghillín breá rása ' teacht faoi ins a' tslí.
Bhí a bhriste sa' bhfaisean 's a watch ar a bhásta,
A chlaidheamh breá faisgthe thairis aniar.
Bhí peara spor geal air den airgead bán,
A hata faoi lása 's a lamhainní buí
Bhí a bhuataisí daite de leathar na Spáine,
A fhuip ina láimh is í teacht mar an Rí.

[coileach]
Casadh mé isteach go teach Thomáis Uí Uada
Bhࢽ cearca go leór ann gan coileach ná ál:
Thosaigh mé orm a' pioca thart thimcheall,
Go dtáinig an neóin gur ghoireas mo sháith,
"Marbhuigh an coileach sin agad, a Mháire,
Is reamhar a chráig is a spruchaille mór."
Tharrainn sí chuicí mé is rinne mo mheádhchain;
Bhí an coisde uilig sásta mo bhruith is mo dhó.

[cearc]
"Mh'och!" ars' an chearc is í ' dul ar a' bhfarra,
"Is brónach 's is deacrach le n-aithris mo sgéal,
Athair mo chloinne 'gus céile mo leaptha
Bheith 'dul ins a' bpota is leac ar a bhíal!
Ta mé 'mo bhaintrigh a' tiocht ráithe an earraigh,
Agus gráinne ní phiocfad dá ngabhfa sa' gcré,
'S is buan tá mo mhallacht tráthnóna 's ar maidin
Do mhnáibh Dhoire Leathain a mharbhuigh mo ghéim!"

A COCK AND A HEN
translation by Donal O'Sullivan

A cock and a hen rambled off with each other;
They travelled all Ireland around and about,
But when they reached Sligo they met with some bother,
so they sent for the lawyers to get them let out,
Before serjeants and judges they took up their station,
But the sheriff stepped forward and gave them a fright,
so they flew to MacNally;s where they got an ovation,
Eating berriew all day, with safe roosting at night.

[hen]
Astride of his racehourse my rooster looked dashing
Amid the great concourse at market or fair,
A watch at his waist and his breeches in fashion,
His gleaming sword drawn for his foes to beware.
His spurs were of silver, a sprig of white heather
Stuck in his laced hat that swept down to each wing,
His shiny boots made of the best Spanish leather,
A whip in his claw as he strode like a king.

[cock]
At the door of Tom Hood's place I stood awhile watching,
There were hens by the dozen but no cock at all;
I strutted around with my picking and scratching,
Till divil a hen but would come at my call,
But the housekeeper caught me and tied me and weighted me,
I felt in my gizzard my end had drawn near;
Beside a sharp knife on the table she laid me,
And that was the finish of poor Chanticleer!"

[hen]
"Alas!" said the hen when she heard the sad story,
"And who is there now to look after my chicks?
To think of an end so untimely and gory,
for my mate on the roost and the father of six!
I'm husbandless now, what's the use of my laying?
Not a grain will I pick till I go to the clay;
But I'll spend all the rest of my days in inveighing
'Gainst the wretches who stole my bold game-cock away"

literal translation

a cock and a hen went off together, And they travelled Ireland till their hearts were broken;
They spent some time in gaol in Sligo, Till they sent word for the lawyers to come. There they were then with bench and bar around them, And it was a senseless person who gave them a fright, And now they are out on grass at William MacNally's, Eating cranberries and sleeping in the heather.

You would have seen my cock on the day of Sraide fair, Astride his fine racing gelding in the roadway, His breeches were in fashion and his watch at his waist, His fine swoord tucked in behind him. He had a pair of shining spurs made of white silver, A laced hat and yellow gloves; His polished boots were of Spanish leather, A whip in his hand as he progressed like the King.

I turned into Thomas Hood's house, There were plenty of hens there, but no cock or brood; I started to pick around and about, Till evening came and I crowed my fill. "Kill that cock of yours, Mary; His claw is fat and his gills are big." She drew me towards her and weighed me; The whole lot of them were content to broil me and roast me.

"Alas!" said the hen as she was going to roost, "Sad and tearful is the tale I have to tell; The father of my children and the partner of my bed Going into the pot with the lid on his beak! I am widowed at the coming of the spring quarter, And not a grain of any sown crop will I pick, And lasting is my curse both morning and evening On the women of Derrylahan who killed my game-cock!"

There are many places named Derrylahan in Ireland, from Derrylahan, Dunamanagh, Cork to Derrylahan, Kilcar, Donegal and several other places in-between. Given the Munster dialect used in this version, it could be the Derrylahan in Co. Cork. On the other hand, maybe it refers to one of several Derrylahans nearer to Sligo. Different placenames are given in the (Connacht?) version in Cas Amhrán; lyrics to follow.