If you chose a transformer / rectifier / capacitor solution, be aware the current in the bridge rectifiers and capacitor is likely to be 2 to 3 times the load current. There are many sites showing the calculations.
Essentially the larger the capacitor, the shorter the time the diodes are conducting and the higher the current.
Question. It is possible you could use the raw rectified voltage to power the cell? you could still PWM the output if you needed variable. And I guess you could use phase control on the input of the transformer.