I found your thread searching for this same story/poem. I called my grandpa who used to tell it to me as a child and he recalled it as best as he could. Hope this helps!

THE MOUNTAINEER AND THE JABBERWOCK
(Phil Harris)

One day there was a mountaineer who felt so mighty brave,
That he took his trusty gun and went to a dark, dark, cave.
The reason for his venture was a story that he'd heard
About a bad ferocious creature called the Jabberwocky bird.

When once inside that cave our hunter saw a sight,
The Jabberwock had seven heads, and each one was a fright.
He turned around and headed home a runnin' all the way,
And to all the folks who waited there the mountaineer did say:

"Twas brillig and the slithy toves did gyre and gimble in the wabe,
All mimsy were the borogoves and the mome raths outgrabe.
O frabjous day! Callooh! Callay! The frumious bandersnatch,
That bird it might have seven heads, but that bird I will catch!"

So once again into that cave the mountaineer did go,
How he's gonna catch that bird he really didn't know.
This time the mountaineer was armed with knives and traps and bait.
He had to catch the jabberwock, or lead a sorry fate.

So once again that ugly bird confronted him once more,
The mountaineer had seven fits and fell right to the floor.
He offered up a hasty prayer and murmured, "it's the end!"
But seven jabberwocky heads said "can't we please be friends?"

The mountaineer just shook his head he couldn't trust his ears,
And seven pairs of jabberwocky eyes were filled with tears.
The Jabberwock broke down and said "a witch made me like this.
And Jabberwocky I must be until a man I kiss."

The mountaineer could not resist the bird's unhappy plea.
He bravely puckered up his lips and said "you'll soon be free."
They kissed and seven heads became just one of golden curls,
The ugly Jabberwock became the prettiest of girls.

'Twas brillig, and the slithy toves did gyre and gimble in the wabe.
All mimsy were the borogoves, and the mome raths outgrabe.
O frabjous day! Callooh! Callay! The frumious bandersnatch
That couple now has seven kids, and that is quite a batch!