Repository: langchain
Issue: ChatLiteLLMRouter: Routing functionality is not working
Labels: 🤖:bug,Ɑ:  core

### Checked other resources

- [X] I added a very descriptive title to this issue.
- [X] I searched the LangChain documentation with the integrated search.
- [X] I used the GitHub search to find a similar question and didn't find it.
- [X] I am sure that this is a bug in LangChain rather than my code.
- [X] The bug is not resolved by updating to the latest stable version of LangChain (or the specific integration package).

### Example Code


```
import os
from litellm import Router
from litellm.router import RetryPolicy, AllowedFailsPolicy, AlertingConfig
from langchain_community.chat_models import ChatLiteLLMRouter
from langchain_core.globals import set_debug
import langchain
import litellm
import boto3
from requests_aws4auth import AWS4Auth


model_list = [
    {
        "model_name": "small",  
        "litellm_params": {
            "model": "bedrock/mistral.mistral-small-2402-v1:0", 
        }
    },
    {
        "model_name": "medium",  
        "litellm_params": {
            "model": "bedrock/mistral.mistral-7b-instruct-v0:2", 
        }
    },
    {
        "model_name": "large",  
        "litellm_params": {
            "model": "bedrock/mistral.mixtral-8x7b-instruct-v0:1",
        }
    }
]

router = Router(model_list=model_list)

chat = ChatLiteLLMRouter( model_name="large",router=router)
chat.invoke("Describe LLM")
```


### Error Message and Stack Trace (if applicable)

AIMessage(content='\nAn LLM, or Master of Laws, is a postgraduate academic degree pursued by those holding a professional law degree, allowing them to specialize in a particular field of law or gain deeper understanding in multiple legal systems.', additional_kwargs={}, response_metadata={'token_usage': Usage(completion_tokens=48, prompt_tokens=14, total_tokens=62, completion_tokens_details=None, prompt_tokens_details=None), **'model_group': 'small', 'model_group_size': 1, 'deployment': 'bedrock/mistral.mistral-small-2402-v1:0',** 'model_info': {'id': 'a97f9695d3aa3def1701731164802172cbc2046d203ed5e1efebdb681bcfd2cc', 'db_model': False}, 'api_base': None, 'caching_groups': None, 'hidden_params': {'custom_llm_provider': 'bedrock', 'region_name': None, 'optional_params': {'temperature': 1.0}, 'model_id': 'a97f9695d3aa3def1701731164802172cbc2046d203ed5e1efebdb681bcfd2cc', 'api_base': None, 'response_cost': 0.000158, 'additional_headers': {}}, 'finish_reason': 'stop'}, id='run-7cf6dba8-919f-4115-a73e-01ee6dffa1bf-0')

### Description

The routing for a specific model is not functioning, by default its taking the first model from the model list.

We have configured three models here, small, medium and large. We have configured ChatLiteLLmRouter to use large, butit used first model in the list, which is small.

In the source code litellm_router.py, its configured to select the first model from the model_list configuration.

**self.model = self.router.model_list[0]["model_name"]**

<img width="996" alt="image" src="https://github.com/user-attachments/assets/c087bbe1-9db8-4a69-90bd-1276d058566b">


### System Info

python -m langchain_core.sys_info


System Information
------------------
> OS:  Darwin
> OS Version:  Darwin Kernel Version 23.6.0: Mon Jul 29 21:14:30 PDT 2024; root:xnu-10063.141.2~1/RELEASE_ARM64_T6000
> Python Version:  3.12.3 (v3.12.3:f6650f9ad7, Apr  9 2024, 08:18:47) [Clang 13.0.0 (clang-1300.0.29.30)]

Package Information
-------------------
> langchain_core: 0.3.15
> langchain: 0.3.7
> langchain_community: 0.3.5
> langsmith: 0.1.136
> langchain_aws: 0.2.3
> langchain_cohere: 0.3.1
> langchain_experimental: 0.3.2
> langchain_openai: 0.2.2
> langchain_text_splitters: 0.3.2

Optional packages not installed
-------------------------------
> langgraph
> langserve

Other Dependencies
------------------
> aiohttp: 3.10.10
> async-timeout: Installed. No version info available.
> boto3: 1.35.45
> cohere: 5.11.1
> dataclasses-json: 0.6.7
> httpx: 0.27.2
> httpx-sse: 0.4.0
> jsonpatch: 1.33
> numpy: 1.26.4
> openai: 1.54.3
> orjson: 3.10.7
> packaging: 24.1
> pandas: 2.2.3
> pydantic: 2.9.2
> pydantic-settings: 2.6.0
> PyYAML: 6.0.2
> requests: 2.32.3
> requests-toolbelt: 1.0.0
> SQLAlchemy: 2.0.35
> tabulate: 0.9.0
> tenacity: 8.5.0
> tiktoken: 0.8.0
> typing-extensions: 4.12.2