Once charges are filed and accepted by the prosecution, the case essentially belongs to them. Legally, the State is the victim of a crime. The person who reported the facts leading to the charge is merely a witness. There is nothing that will absolutely work in order for a complainant to stop the wheels of prosecution once it starts. Ultimately it will be up to the prosecutor.
Most prosecutors will listen to what their victims want to see happen in a case, but they are not bound to do as the victim wishes. It is not uncommon for prosecutors to proceed with a case, especially a domestic abuse case, over the objections of the victim. However, prosecutors also realize that going to trial with an unwilling victim is problematic and will often make an offer that can't be refused to avoid trial or dimiss altogether. A good lawyer will know how far to push the State to get the best result. Regarding whether a victim can file a motion to suppress evidence, the answer is no. The victim is essentially a witness and is not a party to the offense. Having said that, the victim does have rights under Texas law. Google "Texas Crime Victim Bill of Rights" and you'll find plenty of information. Violations of crime victims rights really do not have any consequences, but most prosecutors take the Bill of Rights seriously. Ultimately, the State can force a victim to the stand to testify and, in extreme circumstances, can have a victim arrested and required to post a bond to insure that they will appear when subpoenaed (a material witness bond). The best approach for a victim who does not want to see charges go forward is to make sure that the prosecutors know his or her wishes and to remain consistent. Good luck.
It appears that the victim is at odds with the prosecution. Victim should contact and retain his/her own counsel for advice as to his/her options and the consequences of those options. Neither the prosecution nor the defendant's counsel can advise as they have a conflict of interest.
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.