### 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_
EventPattern=json.dumps(
                {
                    "detail": {
                        "name": [{"prefix": ssm_prefix}],
                        "operation": ["Create", "Update", "Delete", "LabelParameterVersion"],
                    },
                    "detail-type": ["Parameter Store Change"],
                    "source": ["aws.ssm"],
                }
            ),
            State="ENABLED",
            Description="Trigger on SSM parameter changes",
        )
        
        aws_client.events.put_targets(
            Rule=rule_name,
            Targets=[{"Id": target_id, "Arn": queue_arn, "InputPath": "$.detail"}],
        )
        
        aws_client.ssm.put_parameter(Name=f"{ssm_prefix}/test123", Value="value1", Type="String")
        def assert_message():
            resp = aws_client.sqs.receive_message(QueueUrl=queue_url)
            result = resp.get("Messages")
            body = json.loads(result[0]["Body"])
            assert body == {"name": "/test/local/test123", "operation": "Create"}
        
        retry(assert_message, retries=7, sleep=0.3)
        
        self.cleanup(rule_name=rule_name, target_ids=target_id)
    def test_put_event_with_content_base_rule_in_pattern(self, aws_client):