### 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_
)
        invocation_context.context = get_event_request_context(invocation_context)
        relative_path, query_string_params = extract_query_string_params(
            path=invocation_context.path_with_query_string
        )
        try:
            path_params = extract_path_params(
                path=relative_path, extracted_path=invocation_context.resource_path
            )
            invocation_context.path_params = path_params
        except Exception:
            pass
        func_arn = uri
        if ":lambda:path" in uri:
            func_arn = uri.split(":lambda:path")[1].split("functions/")[1].split("/invocations")[0]
        if invocation_context.authorizer_type:
            invocation_context.context["authorizer"] = invocation_context.authorizer_result
        payload = self.request_templates.render(invocation_context)
        result = self.process_apigateway_invocation(
            func_arn=func_arn,
            path=relative_path,
            payload=payload,
            invocation_context=invocation_context,
            query_string_params=query_string_params,
        )
        response = LambdaResponse()
        response.headers.update({"content-type": "application/json"})
        parsed_result = json.loads(str(result or "{}"))
        parsed_result = common.json_safe(parsed_result)
        parsed_result = {} if parsed_result is None else parsed_result