Arduino plus relay is really overkill, just use a simpler method, a small transformer, to say 24VAC, a bridge rectifier, and then a series resistor in the positive output of the rectifier. Then a 1000uF to 2200uF 63V electrolytic (can be any one, 85C is fine here with only 100/120Hz ripple and almost no current in the capacitor) and then a 24VDC relay, that will close to short out a resistor in the feed to the large transformers. Resistor can be any value from around 4R7 to 22R, 10W, with a 130C single shot thermal fuse, held to the body of the resistor with some kapton tape. Thermal fuse on the incoming side before the relay contacts, and the NO contacts of the relay short out the fuse. Relay contacts rated for operating current of the input side, so for a 1kVA transformer on 120VAC mains a 10A contact rating, though a 16A relay is likely to be the same size and price, and available off the shelf in industrial suppliers. Will just last longer, or outlive the equipment.
Series resistor for the relay has to be related to coil resistance, generally around a third to half the resistance, and adequate power dissipation as well. This will take perhaps a half second to close after power is applied, well enough time to limit both initial magnetising current of the toroid, and also charging of the likely massive capacitors on the secondary side. Adjust capacitor value till you get to somewhere between a half second and a second for the relay contacts to close.
Thermal fuse in case the relay contacts do not close and the resistor starts to overheat. All the hot running parts mounted either spaced off the PCB, or mounted using tag strips so they are in free air and not close to anything flammable.
Did that, though I used series resistors to directly rectified mains, as I had some 48V relays with high resistance coils, and used 400V electrolytics instead. No need for a catch diode across the relay coil either, there is no back EMF on it at all, because of the capacitor. Done on a 2kVA variable voltage transformer, that had an annoying ability to trip the thermal breakers used in the supply, despite them being 32A breakers, and not supplying a ring main.