OPAMP saturating when output is negative, simulation is OK
the negative output voltage saturates at -2.95V.


Yup. If you look at Figure 2 of the data sheet, it shows a nominal maximum for +/- 5 volt supplies to be 6 volts pk-pk. In other words, about +/- 3 volts, and you're getting -3. Just about what you'd expect.

The positive side is doing better than expected, but that's just an unexpected bonus.

TL;DR - You should never count on getting +/- 4.1 with your power supply and this op amp.