Hi Hasib, we should phrase it like:
"I would love to be like her, having my son sing at my birthday party."
Here are several different ways of saying it, with small differences.
"I would love to have a son like hers that can sing at my birthday party."
In this sentence, we are focusing on the son's ability.
"I would love to be like her, having a son that can sing at my birthday party."
This one is similar to your sentence, but we are focusing on the difference between our sons.
If we want to focus on the writer, "you", we can say:
"I would love to have my son sing at my birthday party, like her son did at hers.
"I would love to have my son sing at my birthday party, like hers did at her birthday.
These are all different ways of saying it, but I think the closest one to your sentence would be the first one: I would love to be like her, having my son sing at my birthday party."
^ because we are keeping the part of "I would love to be like her".
I hope that helps!