Is it good style to omit the first "that" in the following sentences?
He now knows that the world is his oyster and that he has more options than he would have once believed.
He believes not only that she likes him but also that she will die for him.
Here is a link to a very useful article on whether to omit "that" from a sentence: https://web.ku.edu/~edit/that.html
Without ["that"], the reader's first impression [is wrong]. Remember that even momentary confusion provides readers with a handy place to stop — and that's not good. A reader should never have to pause to understand what the writer is trying to convey. If that happens too often…, a reader stops reading.
For the most part, choosing to include "that" is more of a stylistic choice, as you have noted. In most colloquial settings (e.g. a conversation with a friend), it is perfectly okay to drop the "that", as long as the sentence continues to make sense.
Let's look at your second example:
He believes not only she likes him but also she will die for him.
This sentence is garbled, and doesn't make grammatical sense. Because of this, the "that"s are necessary to include.
In a formal written paper, however, I would suggest including "that" even when it isn't absolutely necessary to make sure that there will be no confusion for your reader.
For example, the beginning of your sentences may have a different meaning than the full sentence, which can cause your reader to have momentary confusion.
He now knows the world…
Does he now somehow know the world in some way?
These are small issues that do not matter much in informal situations. Your first sentence does make sense without "that" in them — it just may be a very slight confusion for your audience. Just be careful in more formal settings, where that confusion could undercut your argument.
