It is not a crime to fail to provide information that graduate programs ask for. However, surely by "illegal" you mean "against the rules of the university and program you're applying to". In that case my answer is no: not sending information to admissions personnel which is a required part of your application and because you think it would jeopardize your admissions chances is both against the rules and clearly unethical. I'm sorry, there's no magic wand to wave to change that situation.
Your hope that with suitable incantations you can perform such an outright academic dishonesty makes me question your academic maturity and suitability for a PhD program. Conversely, if you had those things you might realize that you are being melodramatic and your situation is not as grave as you seem to think.
Five years ago, I obtained a Master's Degree from a top 10 American university in the Social Sciences. I was a novice in academia, only one year post completing my Bachelor's degree, and simply not yet ready for the rigors of professional academia.
I'm sorry, but my eyes are rolling already. When you started a master's degree you'd been in academia for what, 16 years? Hadn't you been taking classes and getting grades for the vast majority of your entire life? In that you had a bachelor's degree, how were you less experienced than anyone else who is starting a master's program? If you were not a serious student during your undergraduate years, you need to take some ownership for that: it's nobody else's fault.
However, I am quite confident that my low GPA of a 3.33 from the first Master's will immediately kill any chance I have of receiving an admissions from American schools,
I'm quite confident that you're painting far too bleak a picture. A 3.33 GPA in a master's program is indeed a lower than average one. But it doesn't kill anything, especially since you got a 4.0 GPA in a different program.
which essentially means that one poor performance five years, at a time when I was an entirely different individual, will forever preclude me from pursuing my life's dreams.
My eyes are really rolling now. What you're saying is both factually incorrect -- no one is ever "forever precluded" from pursuing an academic degree, and your situation is not even an especially tough one to bounce back from -- and really annoying: according to your own description you had more than one poor performance, and you were not "an entirely different individual". Grow up: you had a period of relatively poor (not failing, nothing burned down, you got your degree...) performance. Acknowledge this, and move on.
I do not believe that it is in any way immoral to want send the transcript of my second master's degree, which I feel better represents who I am and what I am capable of, though obviously I am well aware that this is not allowed.
Of course it's not immoral to send the transcript of your second master's degree. You must do that. You must also send the transcript of your first master's degree, which is presumably what you're asking about. And for a third time, my eyes are rolling: doing what is not allowed by a program in order to gain entry to a program is immoral...as you clearly know. You're asking whether there is some penance you can do to nullify this immoral act. Nope.
I am not trying to avoid responsibility for my academic performance,
That is exactly what you're trying to do, or rather asking whether it is possible to try to do.
I am simply looking to pursue my life without being kept back by one poor performance five years ago.
Then get over it. Apply for the programs that you want, at a wide range of schools, and see what happens. Write a personal statement which acknowledges the substandard performance and explains how you've moved on to further success and found clarity in what you want to do. I think it's very likely that you'll get in somewhere. If you don't, you can always enroll in a non-degree program taking individual courses and doing very well in them. Doing that for a few years is the academically acceptable way of laundering your checkered past.