The schematic shows that the bat coil circuit runs off of 120 volts and not the transformer secondary winding.

The schematic doesn’t show a voltage reducing potentiometer, unless one was added.

In coil circuits where multiple inline switches will complete a circuit it’s gonna be the last switch to close that will suffer the arcing.
When a circuit is complete or closed it’s gonna be the first switch to open that will also suffer arcing.
In this case it’s the bat switch that completes the circuit. This switch is probably pitted and needs metal file cleaning.
And on the bat relay a normally closed switch with O-W & BL wires is the switch that opens the bat coil circuit. Clean and adjust switch that the switch opens near full relay pull in.
The bat coil armature switch is a version of a end of stroke switch. When the armature pulls in, a pin on the armature will close the normally open switch, closing the bat relay, opening the circuit to the bat coil even if the bat switch on the top panel is still closed and is the timing of how long the bat coil stays energized.