China already has the production capacity to produce 2.6 billions doses of covid vaccines annually. 500 millions of those doses are from CanSino single dose vaccine. Sinovac has planned to ramp up production to 2 billion doses by June from 1 billion doses. Sinopharm has planned to ramp up production to 3 billion doses within 2 years or so from 1 billion doses. China still has Zhifei vaccine and multiple other vaccines that would be approved before year end. So I think you have underestimated China's ability to meet the demand of vaccines around the world.
At the moment, China doesn't have a supply side problem but a demand side problem. Most people feel relatively safe and have no urgency to take vaccines. That's why they have lowered the goal of inoculating 40% from June to July. Major buyers such as UAE, Turkey, Indonesia, Brazil, and Mexico buy bulk ingredients and do the filling and packaging by themselves. Other small buyers and donations wouldn't be a problem for China.
Siberia has already relied upon Sinopharm for vaccination. Hungary has agreed to procure 5 millions doses from Sinopharm. So it is not unrealistic for Poland or Czech to order 5-10 millions doses and ask to deliver in 4 months. It is a great PR coup for China if Western nations would be forced to rely upon China for vaccines. The naysayers in the third world countries would shut up and allow their government procure Chinese vaccines.
China doesn't need to help everybody. The 5 eyes does not want China's help nor does India, and Chinas own population does not need urgent distribution. The entirety of SEA/LATAM/Africa/MEA can be vaccinated by China alone, especially as China doesnt seem to be going all out on vaccinating its own people (which is medically unnecessary at this point).
The current conservative numbers for FY21 will be:
Sinopharm 1 billion
Sinovac 2 billion
Cansino 500 million
ZF2001 RBD Dimer 300 million
BioNTech - Astra Zeneca chinese production: 300 million
Realistically Sinopharm is a much bigger company than Sinovac and Cansino Biologics combined, and will also hit about 2 billion before end of year. Realistically, baseline production in China is closer to 4-5 billion doses at this stage, assuming none of the other Phase III vaccines get approval.
China needs to be able to dropship 3-5 million doses at once to a country, basically to start the mass vaccination process. If they get doses from Pfizer or COVAX 3-4 months later I'm sure it will be nice but by then people will remember whose plane arrived first and carried the most doses. Also, I think it would be helpful if Sinopharm and Sinovac also shipped into each others customer markets, basically not only to increase depth of distribution (like hundreds of millions of doses to brazil, turkey, and indonesia each) but also to cover breadth of more countries for general awareness.