Reddit reviews Manipulative Monkeys: The Capuchins of Lomas Barbudal

One clue may lie in how Capuchin monkey mothers deal with the deaths of their infants. Capuchin mothers whose infants are killed by males during infanticide will demonstrate hostility and form coalitions against the murdering males. In her book "Manipulative Monkeys," Perry relates how, after Macadamia (infant) was killed, Mani (mom) forms a coalition against Moth (the killer) and a major fight erupts.
>>About a half an hour later, Hannah [a field researcher] calls again: ' She's still got Macadamia. She's starting to handle the body a little more roughly, but she's not ready to give it up yet . She has been threatening and alarm calling at Moth some more.

Some time passes and another infant is thrashed, but lives. Eventually Mani leaves the body and takes off for an hour (Capuchin mothers almost never leave their infants for more than a few seconds) and does not come back.

A year later, Mani had a stillborn infant.
>>Mani was carrying the dead infant everywhere she went... sometime she grasped Pobrecito [the stillborn infant] around the waist, and other times, she held on to the tail or other appendage, letting the body dangle as she locomoted. When she needed to eat or groom someone, she set the body down on a branch, usually continuing to grasp it with a foot. She did, however, occasionally groom the body of her dead baby. Her son Maranon inspected his dead sibling frequently.
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Eventually, the jungle insects do what they evolved to, and begin to eat her dead infant.
>>Around 9:00 AM, flies started buzzing around the body, and this irritated Mani, who kept trying to catch the flies. Mani became preoccupied with the baby's anal region and repeatedly licked and sucked on it, as if ingesting fluids emitted from the body. Although she was concerned with keeping the anogenital region clean, and with keeping the body free of insects, she did not treat the body as if it were alive in other respects. For example, whenever she got a drink of water, she allowed the body to be completely submerged in the water. Mani carried the body all day and slept with it that night.

Mani continued to care for her (now decomposing) infant for another day, but a few times she and others would alarm call at it (as if it was a stranger or a threat), while other times that other females came to inspect it, she'd chase them away. Later that evening, though, she abandoned it too.
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The point I see in both these cases is that the awareness of the mortality of others is the result of maternal caring behaviors, but in the wild, where death meets them regularly, a more complex form of mourning (rumination, depression) would have meant fewer opportunities to eat and reproduce.
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Nature is savage, in order to build a brain that can think of it's own mortality, we'd have to explain why this would benefit an animal, or at least explain why it's a side effect of another behavior that itself is a benefit in evolutionary terms. In an environment where animals rarely make it to old age (though Capuchins actually do make it more often), we'd more often see animals desensitized to it and conversely, if we found animals that evolved in an environment of low mortality (I can't think of any) we might see them evolve awareness of their mortality

tldr: probably not because nature is cruel and death is common.

ps: here is an example of tufted Capuchins, the ones I described.