Question on large/small business certification

The answer depends on how the ISO 9001 certification is structured within the big business.

If the big business is ISO 9001 certified at an organizational level, it keeps certification.
If the certification was contract-specific, the certification may no longer apply to that contract. The small business would need to become ISO 9001 certified if required for the contract.

Well, in this case certification will remain with organisation which is certified and not with your contract or with you. Incase there is any process movement to the small business, it should be included as a part of certified scope or managed as a controlled external provider.