Strange switching behaviour when using bipolar stage to drive mosfets
When you turn the BC547 ON, the low output impedance (emitter follower) quickly charges up the gate and pulls the output low quickly.

When you turn the the BC547 OFF, the gate has to discharge through the resistor network you have which takes some time; the simulator drive is probably set to a very low output impedance and that is why the simulation using direct drive seem to work well.

If you think you need extra gate drive, then either a push-pull arrangement (which actively drives the gate high and low) or a MOSFET driver would work.

You can find an interesting post about it at this location.

To drive a MOSFET you need a low impedance source that can provide the current necessary to charge and discharge the gate quickly; there are some op-amps that can handle it, but check the effective output impedance versus frequency (better amplifiers have it in the characteristics) because the charge current must be provided very quickly. 

In an application driving a transformer, using an op-amp may not be practical due to drain-gate feedthrough.

I would note that LTSpice is supported by Wurth with a full set of transformer models that may let you see the effects more clearly.