I would use a DC SSR with MOSFET output
along these lines. The part number D2D07 should be a suitable choice. I would also incorporate some snubber protection along the lines of the first link I provided. I would use a variac rated for 5 amps off the AC mains and rectify and smooth (filter) the DC out. Then increase the voltage till you achieve the desired current for the buzz coil based on your amp turns of the coil. Build a small astable oscillator around a 555 timer running at your 250 Hz. and use it to drive the SSR. That would be my suggestion or seek your mosfets and build your own SSR design.
Ron