Can I use 2 different nations' passports in China to circumvent visa restrictions?

As asked in the comments, an extension on my original comment, as an answer.

This is the first time I hear of a limit on visas from Hong Kong. My passport is full of them. I'm currently in Shenzhen on a 2-entry visa. When I use both entries, I'll get another one, and another one, and, etc... (NB: that was at the time of the comment. I have used up both entries, and will now get a new one).

Just go back to HK and get a 2-entry visa.

I usually get my 2-entry, 30 days per visit, visas in Hong Kong via a travel agency, one of the big chains. The reason is twofold:

1. The visa section at the China Ressources Building is a circus on the best of days, and the staff there is inflexible and unfriendly. And since France is still on China's shit list, we have to provide a lot more papers than travellers from many other countries. And even if you want to travel to Shenzhen only, they still ask for plane tickets. And travel insurance. Travel agencies get special treatment from the visa section. They have special counters (although in the case of French citizens, again, it doesn't really matter, as we can only get the regular processing time, 5 working days. We cannot get express 1-day visas, so no rush, really...). Their applications are treated much more leniently than individual applications. Mostly because of (2):

2. Travel agencies can produce "real fake" hotel and airplane reservations, proof of insurance, etc... They make real bookings, submit the paperwork, and cancel everything after that. So you show up, fill out a form, give photos, quite a bit of money, your passport, and they do the rest. Works like a charm. It's just quite expensive...