"They want to eat within twenty-two minutes."
Translation:Chtějí jíst do dvaceti dvou minut.
I think there is a difference in meaning of "Chtějí jíst do dvaceti dvou minut." and "Chtějí se najíst do dvaceti dvou minut." The first sentence says that they don't want to wait more than 22 minutes before eating while the second says that they want to be finished eating in 22 minutes.
I think the Czech translation of the English sentence is correct.
If you give such a precise interval specification (22 mins), it's quite understood that you want to be done with the food by the time the interval has passed (because e.g. your train or ferry leaves then). Thus I rest my case that the perfective aspect is more correct than the original.