About the only option you do not want to do is to jump start the car.
The excessive current demands on the charging system of the car might
cause alternator or regulator damage. Some car charging circuits have
protection for excessive battery current draw during charge, most until
recent model years do not.

The best option is to use the lowest current charger until you can
verify 10 volts on the battery unloaded after it sits a few minutes off
the charger. Then any higher current chrage should do fine. This is
only to provide a margin a safety on the excessive current draw the
battery is going to attempt to pull when completely dead. No sense
opening up the thermal fuse inside the transformer in your el-cheapo
battery charger.

I am suprised it actually reads zero volts, unless the battery has an
internal excessive discharge protect cutoff device. Disconnect the
battery and read the unloaded voltage.

FYI, Were it my car with winter coming up, I would not even worry about
it, replace the darn thing. Even the best batteries are less than
$100. It might even be new enough to get a pro-rated warranty
exchange!!!! Having been out in the cold single digits with a bad
battery ONCE, I never chance a weak battery on an upcoming winter.