### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

IaC = CDK.

Attempting to connect an API-GW endpoint directly to EventBridge.

```
      const awsIntegration = new apigateway.AwsIntegration({
      service: 'events',
      action: 'PutEvents',
      integrationHttpMethod: 'POST',
      options: {
        credentialsRole: role,
        requestTemplates: {
          'application/json': requestTemplate,
        },
        passthroughBehavior: apigateway.PassthroughBehavior.NEVER,
        integrationResponses: [
          successResponse,
          serverErrorResponse,
          clientErrorResponse,
        ],
      },
    });
```

Receiving response:

```
    {
      statusCode: 400,
      data: {
        Type: 'User',
        message: `Error invoking integration for API Gateway ID '1z7ky5zzcb': API Gateway AWS integration action URI "arn:aws:apigateway:ap-southeast-2:events:action/PutEvents", method "GET" not yet implemented`,
        __type: 'InvalidRequest'
      }
    }
```

Similar to bug reported:

* https://github.com/localstack/localstack/issues/5508

### Expected Behavior

* 200 response
* Event created in relevant Eventbus

### How are you starting LocalStack?

With a docker-compose file

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

docker run localstack/localstack

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

Calling the API endpoint using axios from jest tests.

### Environment

```markdown
- OS: MacOS
- LocalStack: latest
```


### Anything else?

_No response_
def invoke(self, invocation_context: ApiInvocationContext):
        integration = invocation_context.integration
        integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
        integration_type = integration_type_orig.upper()
        uri = integration.get("uri") or integration.get("integrationUri") or ""
        if uri.endswith("kinesis:action/PutRecord"):
            target = "Kinesis_20131202.PutRecord"
        elif uri.endswith("kinesis:action/PutRecords"):
            target = "Kinesis_20131202.PutRecords"
        elif uri.endswith("kinesis:action/ListStreams"):
            target = "Kinesis_20131202.ListStreams"
        else:
            LOG.info(
                f"Unexpected API Gateway integration URI '{uri}' for integration type {integration_type}",
            )
            target = ""
        try:
            
            
            invocation_context.context = helpers.get_event_request_context(invocation_context)
            invocation_context.stage_variables = helpers.get_stage_variables(invocation_context)