Part of the reason for this sharp change in meaning is that "carry on regardless" is a set phrase. It doesn't include anything about what you might be regardless of, but if it is followed by of that will take precedence. Without that, it's an intransitive phrasal verb with its own meaning, albeit one clearly based on the meaning of the separate words.
So, in "carry on regardless of your opinion", the meaning is like (carry on)(regardless (of your opinion)). Likewise, the second example is (carry on)(regardless (of (if you don't want me to))) - and is somewhat non-natural, possibly even non-standard. More formal and regular would be "carry on regardless of whether you want me to". You could also have "carry on regardless of your not wanting me to" or "carry on regardless of you not wanting me to", depending on dialect.
Your last example, however, is (carry on regardless)(if you don't want me to).
Regardless usually needs an of, for what you have no regard for, but it can get away without it in carry on regardless. Without that, you end up with something that will not be seen as correct:
I will do it regardless of your opinion.
That's fine. Makes perfect sense.
I will do it regardless of if you don't want me to.
That's just as right (and wrong) as your second example.
I will do it regardless if you don't want me to.
This will mostly be see as wrong, but regardless might be taken as an adverb describing the manner in which you will "do it", doing it in a careless way. This is probably a certain degree of meaning being carried back to the word from carry on regardless.