Applying for working visa for other passport

Hi I'm a dual national, let's refer to my passports as country A and country B.

I am scheduled to fly from country A to China in about a month. I will be entering China via visa exemption under bilateral agreement (country B passport).

I was recently offered a job from China. In order to apply for a Z visa for country A's passport, I must legalise my documents and await the PU letter to begin the Z visa process. I can legalise my documents on time but it's unlikely I could get the visa completed within the a month.

I wanted to know if anything could be done to apply for a resident permit/working visa while I am in China under visa exemption from passport B. If not, is delaying my flight and waiting for a visa my only option? Thanks.