I have not filed a 843 so I have no personal experience with it.
However, as you can see at the IRS website, the form is only one page and not complex.
The most important thing (to me) is that this means that you are telling the IRS that this "employer" is refusing to refund the excess SS tax withheld. This should result in either an abatement to you from the IRS, or the IRS writing a letter to your employer asking them to refund the tax, or else. I am guessing that most employers would like to avoid IRS letters.
In fact, it might help if you asked your PEO for a letter explaining their position and attach it to your request for abatement. Since you are going to name the company anyway, it might motivate them to work with you.
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