When you use a quotative-と, the "quote" will basically be a normal non-polite sentence. That is, you can safely use は inside the quote, and you should not drop だ. Brackets are usually not used in a simple case like this. Therefore, the correct sentence is:
The polite version is:
As you may already know, 私は is normally ...
Perhaps it would help to rearrange it into a more standard word order?
This と is just the standard quotative particle associated with 約束する.
In the lyrics she just said 約束する first and followed that with the content of the promise.
You mention that you want to focus on speech. In speech, I can't recall ever having heard anyone use である in a non-joking sense unless it was in a 時代劇. We can and do say "ではある" to provide contrast or affirmation (see here), but that wouldn't be used in this kind of construction.
With that out of the way, let's first look at the nominal connectors you've ...