How can I apply inversions to sentences that include "because"?

TL;DR

In short, you don't. The word "did" is used to modify the other verb, it is included in the question form, but not included in the statement form.

Pair the statement with the question

I think it is helpful to build context around the "because" statements that you provided. Usually subject-auxiliary inversion is used with questions and answers. It can be helpful to think about the questions and answers that pair together.

The question paired with the statements you provided ("Because did I love the dinner, I ate it to my full" and "Because I loved the dinner, did I ate it to my full") would be:

Why did you eat the dinner until you were full?

With this being the question, here are a couple of examples of well formed statements in response:

I ate it until I was full, because I loved the dinner

and,

Because I loved the dinner, I ate it until I was full.

Note that the latter statement will sound slightly awkward to native speakers, but it is still correct

Subtle complexity in the content clause

The part that may be tripping you up is known as the penthouse principle in relation to content clauses. I'll try my best to explain it in the context of your question.

You may have noticed the word "did" is not present in the statement form. This can be very confusing, but a way that helps me think about it is this. "Did" in the question, "Why did..." is not the important verb. The verb (or action) in question, comes after that. In the case of "Why did you eat", the verb in the question is "eat," not "did." When answering a question like this, you do not include the verb "did" as it is only relevant to the tense of the other verb. "Did" only informs you to use the past tense of "eat" like you have noted in your example.

"Why did you eat" --> "I ate, because..."

"Why do you eat" --> "I eat, because..."

This was a long answer to a short question, "How do I apply inversion to a sentence that includes 'because'?"

In short, you don't. The word "did" is used to modify the other verb, it is included in the question form, but not included in the statement form.

Appendix

You will see this take form in other questions, "Why did..." or "How come...", in all cases do not include the verbs "did" and "come" in your answer.