The hyperfoam model in Abaqus is very similar to the standard hyperelastic Ogden model. The reason I sometimes recommend the hyperfoam model for foam materials is simply based on experience. I have used it multiple times and it works relatively well for low density soft polymer foams.
Besides that, there is really only one reason it migth work better than the other hyperelastic material models. The reason is that it has the ability to specify the volumetric response a sum of different terms. This can potentially improve the predictions.
You are also right about the FeFP model - it's not my favorite at this time.
It should be possible to accurately predict the experimental data that you provided using the Micromechanical Foam Model (MFM), and yes, the MFM can simulate plasticity.
Jorgen Bergstrom, Ph.D.