### 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_
integration_responses = integration.get("integrationResponses")
        if not integration_responses:
            return response._content
        
        
        integration_status_codes = [str(code) for code in list(integration_responses.keys())]
        
        
        if status_code not in integration_status_codes or not integration_status_codes:
            return response._content
        
        
        selected_integration_response = select_integration_response(status_code, api_context)
        response_templates = selected_integration_response.get("responseTemplates", {})
        
        
        
        
        template = response_templates.get(APPLICATION_JSON, {})
        if not template:
            return response._content
        
        variables = self.build_variables_mapping(api_context)
        rendered_tpl = self.render_vtl(template, variables=variables)
        
        response._content = json.dumps(ast.literal_eval(rendered_tpl.strip()))
        LOG.info("Endpoint response body after transformations:\n%s", response._content)
        return response._content