Steve -- I've been singing that alternate version for about 30 years. Heard it on the radio -- Maybe from "The Albion Band".
Slight difference in the words. It begins each verse with similar words to the version above but drops the second part. Also slight changes in the refrain from your one. Takes a while to sing but it makes a good group song.
Here it is (phonetically rendered):
The Fox (alternate version)

1. The fox went out on a windy night
With a ling dong derry dong kyro me
Prayed to the moon to give him light
With a ling dong derry dong kyro me
Hey fo lal lee fol lar fol la lair-o
Hey fol la lair fol la lair-o me
Up jumped John ringing on his bell
With a ling dong derry dong kyro me.

2. He ran 'till he came to a great big bin
With a etc.
Where the ducks and the geese were kept there in
With a etc.
Hey etc.

3. He grabbed the grey goose by the neck.
With a etc
Threw a little duck right across his back.
With a etc
Hey etc.

4. Old Mother Slipper Slopper jumped out of bed.
With a etc.
Out of the window she popped her little head.
With a etc.
Hey etc.

5. John he ran to the top of the hill.
With a etc.
Blew on his horn both loud and shrill.
With a etc.
Hey etc.

6. Fox he ran to his own den.
With a etc.
There were the little ones eight nine ten.
With a etc.
Hey etc.

7. The fox and his wife without any strife.
With a etc.
Cut up the goose with a fork and a knife.
With a etc.
Hey etc.