I don't think the 2nd sounds natural in this case. The other two are fine.
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RegardsYour sentences are too formal and not likely to be said in a casual conversation." Lisa asked her mother if she could have a pet ....". Why would this be unnatural or not likely used for conversational English.
RegardsWhy do you think they are better? Why don't you answer
hrsaneiI don't want to stoke the argument by taking sides.Hi hrsanei,
But dimsumexpress is right about his suggestions. His comment was useful to the original poster.
hrsaneiLisa asked her mother to let her keep a pet, but her mother