### 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
        uri = integration.get("uri") or integration.get("integrationUri") or ""
        template = integration["requestTemplates"].get(APPLICATION_JSON)
        account_id, queue = uri.split("/")[-2:]
        region_name = uri.split(":")[3]
        if "GetQueueUrl" in template or "CreateQueue" in template:
            request_templates = RequestTemplates()
            payload = request_templates.render(invocation_context)
            new_request = f"{payload}&QueueName={queue}"
        else:
            request_templates = RequestTemplates()
            payload = request_templates.render(invocation_context)
            queue_url = f"{config.get_edge_url()}/{account_id}/{queue}"
            new_request = f"{payload}&QueueUrl={queue_url}"
        
        headers = get_internal_mocked_headers(
            service_name="sqs",
            region_name=region_name,
            role_arn=invocation_context.integration.get("credentials"),
            source_arn=get_source_arn(invocation_context),
        )
        url = urljoin(config.service_url("sqs"), f"{get_aws_account_id()}/{queue}")
        result = common.make_http_request(url, method="POST", headers=headers, data=new_request)
        return result
class SNSIntegration(BackendIntegration):