Just because the I-9 isn't submitted to the government doesn't mean that this isn't a problem - it still very much is.
Since 1996 when a new law on this became effective, any false claim made for any purpose or benefit to anyone - not just to a government official - makes a person inadmissible. Permanently. With no waiver available.
I would speak with an attorney to examine your case in-depth.
The Code section listing "excludable aliens is 8 U.S.C. § 1182
which is available at:
---Start of Excerpt--
(ii) Falsely claiming citizenship
(I) In general
Any alien who falsely represents, or has falsely
represented, himself or herself to be a citizen of the
United States for any purpose or benefit under this chapter
(including section 1324a of this title) or any other
Federal or State law is inadmissible.
---End of Excerpt--
At the top of the web page you see the chapter is "Chapter 12" and when you click on that "Chapter 12"
But also note specifically that the section mentioned (1324a) deals with employment and is the basis of the I-9 form.
Therefore, you would have a big problem getting your green card if the USCIS found out about your false claim of citizenship. Idon't know what you "false claim" was and that might be relevant. I would be willing to work on your matter further , but I would need to spend a substantial amount of time and that would be beyond the scope of this free forum. If you want to inquire about hiring me for such work, please contact me directly to the email address below.
Thank you for asking,
Harvey Mechanic, Attorney at Law
Authorized to handle Immigration matters throughout the United States