"A generator is swell, but there's a cheaper option-- a battery backed
secondary sump pump. "

Your comparing apples and oranges. A battery backup unit is great for
a short interruption in power. The OP was asking about protection from
power outages during a storm. IF you have a significant storm that
puts out the power, there is a very good chance that the power won't be
back on before the batteries are dead and the basement is flooded.

The battery backup together with having a manual transfer generator
can be a viable option. The batteries will give you time to get home
to start the generator. But If you have city water, one of the water
driven backup pumps is superior to all of this, if the sump pump is the
only power outage issue.