### 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_
"Key": "overwritten_query",
                            "Value": "original",
                            "IsValueSecret": False,
                        },
                    ],
                },
            },
        )["ConnectionArn"]
        
        dest_name = f"d-{short_uid()}"
        result = aws_client.events.create_api_destination(
            Name=dest_name,
            ConnectionArn=connection_arn,
            InvocationEndpoint=http_endpoint,
            HttpMethod="POST",
        )
        
        rule_name = f"r-{short_uid()}"
        target_id = f"target-{short_uid()}"
        pattern = json.dumps({"source": ["source-123"], "detail-type": ["type-123"]})
        aws_client.events.put_rule(Name=rule_name, EventPattern=pattern)
        aws_client.events.put_targets(
            Rule=rule_name,
            Targets=[
                {
                    "Id": target_id,
                    "Arn": result["ApiDestinationArn"],
                    "Input": '{"target_value":"value"}',
                    "HttpParameters": {
                        "PathParameterValues": ["target_path"],