The "Vive La/ Clare's Dragoon's" tune clearly derives from a very similar tune called "Willy was a Wanton Wag."

The melody of "Will..." (sic) appeared in "Aria di Camera...A Choice Collection of Scotch, Irish & Welsh Airs for the Violin and German Flute by the following masters: Mr. Alex. Urquahart of Edinburgh, Mr. Dermot O'Connar [sic] of Limrick [sic], Mr. Hugh Edwards of Carmarthen” (London: Dan. Wright, [ca1727]).

One presumes, then, that the tune was composed by Alexander Urquahart, who has not been certainly identified.

The words of "Willy was a Wanton Wag" likewise date from 1727, appearing in "A New Miscellany of Songs" (London: A. Moore), pp. 202-203.

Haydn published an arrangement of "Willie..." (sic), with the words, in the 1790s.

The Irish collector George Petrie (d. 1866) knew the tune (in "The Complete Petrie Collection") as "Vive La, or The French are Coming."

For various versions of the tune, which was evidently very popular in the 18th century, see the Traditional Tune Archive, which (very) eventually led me to this information,

The 1727 lyrics:

Willy was a wanton Wag,
The blythest Lad that e'er I saw,
At bridals still he bore the Brag,
And carried aye the Gree awa.
His Doublet was of Zetland Shag,
And wow but Willy he was braw.
And at his Shoulder hang a Tag
That pleas'd the Lasses best of a'.

He was a Man without a Clag,
His heart was frank without a Flaw,
And ay whatever Willy said,
It was still haden as a Law.
His Boots they were made of the Jag,
When he went to the Weapon-Shaw,
Upon the Green nane durst him brag,
The Fiend a ane amang them a'.

And was not Willy well worth Gowd?
He wan the Love of Great and Sma';
For after he the Bride had kiss'd,
He kiss'd the Lasses hale-fal a';
Sae merrily round the Ring they row'd,
When be the Hand he led them a',
And Smack on Smack on them bestow'd,
By Virtue of a standing Law.

And was na Willy a great Loun,
As shyre a Lick as e'er was seen?
When he danc'd with the Lasses round,
The Bridegroom speer'd where he had been.
Quoth Willy, I've been at the Ring,
With bobbing, Faith my Shanks are Sair;
Gae ca' your Bride and Maidens in
For Willy he dow do nae mair.

Then rest ye, Willy, I'll gae out,
And for a wee fill up the Ring,
But, Shame light on his souple Snout,
He wanted Willy's wanton Fling.
Then straight he to the Bride did fare,
Says, Well's me on thy bony Face,
With bobbing Willy's Shanks are Sair,
And I am come to fill his Place.

Bridegroom, she says, you'll spoil the Dance,
And at the Ring you'll ay be lag,
Unless like Willy ye advance;
(O! Willy has a wanton Leg)
For wi't he learns us a' to steer,
And foremast ay bears up the Ring;
We will find nae sic dancing here,
If we want Willy's wanton Fling.