No, we do not use 'reason why'; many grammarians consider it poor style. We use 'reason that' to be safe. And 'why' certainly does not work with a participle ('being') anyway.
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Mister Micawber'reason..because' and 'reason...why' are both considered poor style by someWhile I don't disagree, it's instructive to Google "reason why grammar". You'll get an amazing number of hits on sites which purport to teach proper grammar.