Call Bradford White customer service and ask THEM.

They have a toll free number and speak English.

Have the model # of your WH handy.

"What are the benefits for having electrical ignition?"

Is this just a standard water heater with a central flue or is there a controlled draft? No pilot means less gas consumption at minimum. If there is controlled draft, more efficiency. If there is controlled draft you probably are not going to fire it up safely without electricity.

"What are the benefits for having electrical ignition?"

No hot water if you do not have power.

It should shorten showers considerably.

The pilot is not enough gas to worry about, and the efficiency (or lack) is mainly driven by the (poor) heat exchangers in the typical tank unit.
A spiral of sheet metal up the in tank flue line.

I would be willing to bet it's a power vent model. No power would mean the exhaust gases would leak back into the house. Very dangerous.

A power vent will have a plastic exhaust vent, and an electric fan unit on top of the heater. If it's really a b-vent type, with a metal 'chimney' with an upside down funnel arrangement at the bottom above the heater, it could run without power provided you can solve the ignition issue.

Of course, modifying such a unit may not be permissible under codes.