I think the OP's wife ran into a very unique situation. As far as I know, one never had to get permission to leave Hong Kong. My children were born in HK and immigrated at a very young age and have applied for China visas numerous times with their U.S. passports and were never asked to provide proof that they left Hong Kong "with permission". In fact, I looked at their old HK passports and there weren't even any HK exit stamps in them. I can understand the Chinese Embassy asking for proof if the child was born in China but Hong Kong? Besides, the visa application does not even ask what age you left your former country so I assume the question came up when the OP's wife was applying for the visa personally at the Embassy. I suggest the OP try using a Chinese Visa service in another city.