I suggest two plans, each of which I have been able to implement in a partial way only:
1. Take the train around to random first, second, and third tier Chinese cities. Many of them will have their own cuisines, or they will represent a nearby regional cuisine. It’s like discovering the food of a new country. Imagine if Shandong province were a separate country! How compelled you would feel to visit it for the food, often considered China’s foundational cuisine, plus it uses the finest vinegars. And yet, because it is part of “China” (Gavagai!), you feel you already know something about Chinese food and thus the need to sample it is not so pressing. Redo your framing, and rush to some of the lesser visited parts of China.
By the way, you can stay in the second or third best hotel in most Chinese cities for only slightly more than $100 a night, and yet receive five star treatment and quality.
2. How many provinces does China actually have? I don’t wish to litigate that dispute, but most of them have restaurants devoted to their regional dishes in Beijing. These are state-owned restaurants, and most of them are excellent. Furthermore they are scattered around town, so if you visit them all you will see many parts of Beijing.
A month in Beijing should allow you to visit them all, plus the air pollution really is better these days.
I should add that western China has by far the best raisins I have sampled in my life, most of all the big red raisins. Until my trip to Xi’an, I had never actually tried a real raisin with the real raisin flavor. Forget the Terra Cotta Warriors, discover what a raisin is!