Yes, if it's rectified, unfiltered DC, then that could cause a problem, when the voltage is low, since the MOSFET won't turn on properly, although the current will also be low at this point, so the power dissipation might not be too high.
Why not simply use a 24VDC switched mode power supply, rather than a transformer, to power the whole circuit? They're not expensive and using DC, rather than AC, will make things easier.