Can I sue the company if I have evidence that the document which the PIP was based on was based on lies?
An aggrieved person can sue almost anyone for any reason.
The question you must ask yourself is: "If I bring a lawsuit against "Company XYZ" or "Mr Bigshot" will I prevail?
What would be my remedies?
If you plan to sue your employer for a performance review you dislike, or for placing you on a "PIP", prepare to be terminated. If you initiate such a lawsuit, prepare for your employer to prevail in court.
Texas is an "at will" employment state.
If you are employed at will, your employer does not need good cause to fire you.
In every state, but Montana (which protects employees who have completed an initial "probationary period" from being fired without cause), employers are free to adopt at-will employment policies, and many of them have. In fact, unless your employer gives some clear indication that it will only fire employees for good cause, the law presumes that you are employed at will.
Even as an at-will employee, you still cannot be fired for reasons that are illegal under state and federal law.
In the above instance, our laws make an exception to the general rule of at-will employment.
If I cannot sue what would be the optimal and most efficient move I can make that can get me as close to keeping the job as possible?
To complete the PIP and work as hard as humanly possible to get back in your manager's good graces.
Your other option is to seek new employment before things go from bad to worse.