Calling Cognito Sign in from Lambda causes Timeout

I am implementing user login using AWS cognito user pool. So the flow is, Website will call an API Gateway, which will trigger a lambda function (node.js 18.x), which in turn will interact with Cognito user pool through the 'amazon-cognito-identity-js'. The node.js code authenticates user within a second in my local, but when I deploy it in lambda and run it there, it takes an excruciating 10 seconds to authenticate(Had to increase the timeout from the default 3 seconds). The sign-up, email verification flow, they work within a second in lambda, but I am facing issues with sign-in for now, haven't tried the forgot password flow yet. The lambda sign-in code is as follows:

import {CognitoUserPool,AuthenticationDetails,CognitoUser, CognitoUserAttribute} from 'amazon-cognito-identity-js';
import dotEnv from 'dotenv';

dotEnv.config();




const signIn=(cognitoUser,authDet)=>{
 return new Promise((resolve,reject)=>{
     cognitoUser.authenticateUser(authDet,{
         onSuccess:(session)=>{
             console.log(`Success: ${JSON.stringify(session)}`);
             resolve(session);
         },
         onFailure:(err)=>{
             console.log(`Failure: ${JSON.stringify(err)}`);
             reject(err);
         }
     })
 });
}

export const handler= async (event)=>{
 console.log(`The event: ${JSON.stringify(event)}`);
 const poolData={
     UserPoolId:process.env.COGNITO_POOL_ID,
     ClientId:process.env.CLIENT_ID
 }

 const userPool=new CognitoUserPool(poolData);
 let response={};

     console.log('Signing In');
     const requestBody=JSON.parse(event.body);
     const authDet=new AuthenticationDetails({
         Username:requestBody.email,
         Password:requestBody.password
     });

     const userData={
         Username:authDet.username,
         Pool:userPool
     }

     const cognitoUser=new CognitoUser(userData);
     try{
         const session=await signIn(cognitoUser,authDet);
         response={statusCode:200,
             headers:{'Content-Type':'application/json'},
             body:{state:"Success",
             session:session}
         }
     }catch(err){
         response={statusCode:200,
             headers:{'Content-Type':'application/json'},
             body:{state:"Failure",
             err:err}
         }
     }
     console.log(`Response to return: ${JSON.stringify(response)}`);
     return response;
 
 
   
}

The lambda configuration is the default one, with no VPC, or any other special network config. I just created a lambda function and deployed this node.js into it. Both the user pool and lambda are in the same region. The API gateway has a proxy integration with lambda.

I know AWS documentation talks about doing it the other way round, i.e, calling cognito directly from website and have post-confirmation and pre-token generation triggers. But, that way the pool id and client id remains exposed on client machine, so I thought of doing it this way. It baffles me why sign in is taking such a long time when done through lambda, but when I do it from website (or from local), it authenticates within a second or two.

If anyone could help me out in figuring out why the timeout occurs, or if there is something wrong with my code, then that would be great.

Thanks...