The remedy for failure to provide the Miranda warnings is they the police cannot use your statement against you in court. You need an criminal defense attorney. Your attorney can work to reduce the impact of this charge on your life. Time to switch from thinking about suing the police (not viable here), to protecting your future. MAKE NO admissions here on this public forum.
We do not have a client/attorney relationship until you make an appointment, we discuss your case face to face, I accept a retainer, and we explictly agree to enter into representation.
If you were not questioned/interrogated in a custodial setting, cops do not have t read you your rights. You cannot be interrogated without your attorney if you demand to have him or her there.
You need to hire an attorney. He or she will know how to address the actions of the cops.
The above is not intended as legal advice. The response does not constitute the creation of an attorney client relationship as this forum does not provide for a confidential communication.
It is impossible to predict how difficult it will be to, in your words, beat this false charge. I say that because there are many variables (witness statements, photographic or other physical evidence, etc.) that can influence the outcome which should not be discussed in public. You should consult privately with several defense attorneys and try to find one you trust to represent you.
Best of luck,