I can’t offer any advice on getting whatever visa it is you’re trying to get, because the visa environment in China has changed recently, so I’m not familiar with what is possible now.
I would say to not blindly trust some Chinese visa guy. In my experience, it’s never as quick and easy as they make it out to be. I’m sure there are services out there that can figure it out, but you need to be diligent and really make sure you have all the required documents and have backup plans.
After you leave China you can immediately re-enter on the same day. As long as you get that exit stamp, your next entry stamp renews your stay period. You don’t need to talk to anyone for this. Just pass through immigration both ways. Finding same day flights that provide enough time for you to do this can be more problematic, but people do this through HK all the time.