How does the voltage across a battery compare to the voltage across the light bulb?

Without a circuit diagram, there are more questions before answers.

First, assuming no other components, there is wire resistance to consider. There is also minor resistance in each connection, and each switch contact.

In any event, the sum of the voltage drops in any series ckt equals the voltage of the source. If the sum of the wire, contact, and other voltage drops was .2 volts, and the source was 12.6 volts, then the light bulb would have 12.4 volts applied to it.