Negative volume cells is typically due to unreasonable displacement received by Fluent or unreasonable forces sent to Mechanical. Assuming your mesh is of good quality, since you are following Best Practices recommendations from our user guides, it is a good idead to monitor the pressure/force convergence and displacement convergence for those FSI surfaces within a time step to see which of the above is causing problems.
Another thing to look at are the Solver Controls for Mesh Displacement equations. In the Convergence Criteria, you may try to decrease the Residual Target and and Increase the Max Coeff. Loops.
I hope this helps.