Nuair a bhí an béile ite.... what is this ite, and why not ithe? I am very confused here, can someone help?

It's the verbal adjective, eaten.

Ohhh but what about ithe?

I am probably getting two things very mixed up

Not very. 'Ithe' is the verb as well as the verbal noun in this case.

Try this.

http://www.csis.ul.ie/focloir

It's an Irish/Irish dictionary. Just type in the word you need and click "cuardaigh".

ith = eat!
ithe = verbal noun "eating"
ite (= ith + -te) = verbal adjective "eaten"
compare with buailte = buail + -te = "beaten"
You do not write "tht" (*ithte) but simple "t" (ite).
It's an orthographic convention.

GRMA a Lars, that makes it very clear to me!