### 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_
if response and response.content is not None:
        response.headers["Content-Length"] = str(len(response.content))
def apply_response_parameters(invocation_context: ApiInvocationContext):
    response = invocation_context.response
    integration = invocation_context.integration
    int_responses = integration.get("integrationResponses") or {}
    if not int_responses:
        return response
    integration_type_orig = integration.get("type") or integration.get("integrationType") or ""
    integration_type = integration_type_orig.upper()
    if integration_type == "AWS_PROXY":
        LOG.warning("AWS_PROXY integration type should not apply response parameters")
        return response
    return_code = str(response.status_code)
    
    selected_integration_response = select_integration_response(return_code, invocation_context)
    
    response.status_code = selected_integration_response["statusCode"]
    response_params = selected_integration_response.get("responseParameters") or {}
    for key, value in response_params.items():
        
        if str(key).lower().startswith("method.response.header."):
            header_name = key[len("method.response.header.") :]
            response.headers[header_name] = value.strip("'")
    return response
def invoke_rest_api_from_request(invocation_context: ApiInvocationContext):