"Looking forward to(something)" is a phrasal verb, meaning that you treat the phrase as you would a verb. Therefore, the something in question can only be either a
So, to eat+ing is the correct answer.
“Looking forward to eating” is correct here.
“Looking forward to” is acting like a phrasal verb that wants a direct object — a noun.
I am looking forward to the test
She is looking forward to the new “Star Wars” movie.
They are looking forward to a better future.
We are looking forward to running a marathon.
I was looking ...
Option #1 is grammatically correct, and sounds natural to my (American) ear.
Option #2 is grammatically correct, but does not sound natural to me. It has the same meaning as Option #1. It has the same level of formality as Option #1.
Option #3 is not grammatically correct.