'Without a battery' is accepted.
'Out of battery' is incorrect because the battery is a device not a fuel. The term to use when you have a flat/dead battery in Italy is una batteria scarica/a terra. The latter is also used for a flat tyre, but note the difference in the nouns: una ruota a terra, o una gomma sgonfia, i.e. the wheel is a terra and the tyre is sgonfia, which makes good sense.
Yes you would, neither of the options given sound like native English. The most natural translations depending on the intended meaning would either be "the car is out of battery" if referring to the charge in the battery or "the car doesn't have a battery" if the car is missing it's battery.
There are many ways of translating a sentence like this one. Even though the meaning of the whole sentence is the same I think "lacks battery" is not the most straightforward translation. Keep in mind they have a limited amount of translations they can support for each sentence and they should support the most frequently used ones.
You can say "that car has no battery" or "that car is without battery" if you mean that has no battery device.
You can say "that car is out of battery" if you mean it does have the device, but it's out of charge.
Duolingo wants us to understand the first situation.
But since the word "battery" carries both the meanings (device and charge), we could not guess without a context, so both could be accepted. Because native speakers actually would use any of these sentences to express any of the meanings in daily speaking.
But it's actually appropriate that duolingo only accepts one of the interpretations, otherwise we would not be discussing it here, and we would miss the opportunity to be aware of this ambiguity.
Even though, the most common situation is the second one, (out of charge). Which most of the people would understand this way, not so often cars get out of battery devices as they do get out of charge. So maybe, Duo should consider this one instead.
But again, less people would be discussing it. So whatever, I liked this one!
Anything to add, please do. Thx!
While I read the direct translation as "That car is without battery." As an American I'd never say that. I say "That car doesn't have a battery." Meaning, it won't run because if I open the hood I won't see a battery in there. I think of it like cars don't actually "have" batteries in the sense of ownership as we think of it. But neither do Italians "have" years, or hunger, or thirst, or fear, or need, or days.
I have read a few comments still troubled though. Here's what I put and still got it wrong, "that car is without a battery". Now I thought that was a pretty straight translation, and while the English grammar maybe a bit old-fashioned , I was surprised it was marked wrong as it seemed to get the gist of the interpretation as well.