Answer - C.
The best way is to create a separate stage in the API gateway as maybe ‘v2', and then customers could use both API versions.
They can still slowly change their usage onto the new version in this duration.
Below is the concept of the API stage in the AWS Documentation.
API stage.
"A logical reference to a lifecycle state of your API (for example, 'dev', 'prod', 'beta', 'v2')
API stages are identified by API ID and stage name".
Options A and B are incorrect since access needs to be provided via the gateway.
Option D is incorrect since you need to keep both versions running side by side.