Past perfect- Past simple

There are two sentences. The first expresses the fact and the second gives the reason.
Yes, your changing of the tense is OK.
A third way might be:
When I arrived home from school, I wasn’t hungry. I had eaten on my way home.
or
Upon coming home from school, I found I was not hungry; because I already ate on the way.

LeGion12359I wasn't hungry when I came home from school. I already ate on my way home from school?

That one is not natural in BrE. We would use the past perfect there.

wilpeterWhen I arrived home from school, I wasn’t hungry. I had eaten on my way home.

This is better because, the word "school" was used more than once in the question and was therefore, repetitive.

wilpeter because I already ate on the way.

Although I listed it as a suggestion, I didn't weight the choices. I personally avoid using "ate" by itself in a sentence; however, I do sometime use "because I already ate dinner on the way."

wilpeterI personally avoid using "ate" by itself in a sentence; however, I do sometime use "because I already ate dinner on the way."

My understanding is that this is perfectly natural in NAmE.