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“my kid” is more natural sounding and indicates parenthood a bit more obviously.
“My boy” is perfectly acceptable too. Might be a regional thing but it does exactly what you said.
Reuse of the punch line within the setup diminishes the tension. Subsequently the payoff is less effective.
The punch line is that “son” and “sun” sound the same. “my kid” isn’t a reuse of the punchline any more than “my boy” is, but it reads less awkwardly.