That won't work at all.These 2 R-C combinations might be replaceable by simple diodes (cathode at 12 V, anode at the end of the coil) - you can try to put the SBLs there.
I'd expect this to work like a charm.
One MOSFET drain will be at 24V whilst the other is switched on, simply by the transformer action of the two parts of the primary winding.
In a less-than-perfect transformer there will be some overshoot.