The title question does not ask about the resultative adjectival usage 'grow calm' but about the variant 'grow to be calm' and the verb usage 'calm'.
- Pups that were fawned over grew [up] to be calm regardless of their natural mother's behaviour.
is idiomatic (especially with the 'up') with the intended meaning.
- Pups that were fawned over grew calm regardless of their natural mother's behaviour.
does not work, as it implies a relatively fast process, not the temperament modelling @Kate mentions.
- Pups that were fawned over were calm regardless of their natural mother's behaviour.
works, but with a different (depictive not resultative) meaning.
- Pups that were fawned over calmed regardless of their natural mother's behaviour.
is probably unidiomatic, though adding padding (quickly calmed or calmed down [quickly]) forces the intransitive usage of the verb 'calm' / 'calm down':
calm [verb] [a intransitive verb]
:to become calm — usually used with down
- The mayor asked the protesters to calm down so he could speak.
[Merriam-Webster]
(fuller ELU-directed answer)