She is not in the process of becoming good or in the process of getting good, she already is being good. The writer has no doubts she shows a bright side. When does she show a bright side? Anytime when she is being good to everyone. So 'when being good to everyone,' is correct.

It is the placement of the commas that make the sentence perfectly acceptable, but a little outdated in its style. "anytime" has been marked out from the rest of the sentence. To mean "I don't mind her showing her bright side anytime, she is being good to someone" The sentence is a lit bit sarcastic and suggestive of her only being nice to someone when she is seeking a reward of some kind. My alternative sentence is more in keeping with how we would say it currently today. "I don't mind her showing her bright side anytime, when/whenever is being good to someone" Yes you can use when and being together and it is not becoming as the sentence is written. The punctuation is a little bit awkward, I think even my sentence may be better as a complete longer sentence without any punctuation. .

@Gary _ What a relief! Thank you.

As John said, it's fine. 'becoming' and 'getting' are not very natural with 'good' anyway.