Good quality copper 15AWG wire is supposed to be 0.01 ohms/m. Therefore the O.P's coils will have a total resistance of about 0.3 ohms each. In series that's 0.6 ohms. O.P.'s quoted 0.34 ohms so either its crappy wire or he cant measure low ohms accurately.
With an iron path length of 400mm, and just about any Iron alloy having a relative permeability greater than 1000, the reluctance of the air gap will certainly dominate if its over 0.4mm and may dominate right down to 0.1mm. Its therefore probable that the only reluctance that needs to be considered is that of the air gap and the pickup magnet, and the only saturation flux that needs to be considered is that of the magnet which will need to be taken well into saturation to remagnetise it.
The core circumference is 0.22m so with a total of 60m of wire its impossible to get more than 273 turns. If the coils are scramble wound it could be much less. Is approx 250 Ampere Turns going to be enough to remagnetise the O.P's pickups at a current the coils can carry without burning out
Its fairly obvious that rectifier losses will dominate, and all that series resistor does is get hot and make you need a larger transformer. Its only use is if the transformer output voltage isfar greater than required to drive the required current through the bridge rectifier and work coils.
Rather than applying continuous DC which will cook the coils due to I2R losses if you want high flux, it may well be preferable to shunt the coils with a really high current Schottky diode, cathode positive to prevent resonant reversal of the current and pulse discharge a large capacitor bank through them. You could easily put a peak current of hundreds of amps through the coils that way. If you use a mechanical switch it would need to either be very heavy duty or to have replaceable contacts due to arcing during the contact bounce period trying to weld them together. A high current SCR would work if the gate drive is hard enough - you woud want to drive the gate with approx 80% of the max gate current rating. If the SCR blows (shorts), the peak current was too high so either get a beefier one or decrease the voltage you charge the cap bank to.