Custom `user-agent` not propagated to nested STS client

[jackrein]

Checkboxes for prior research

• I've gone through Developer Guide and API reference
• I've checked AWS Forums and StackOverflow.
• I've searched for previous similar issues and didn't find any solution.

Describe the bug

When using role chaining, the custom user agent on the outer client is not propagated to the nested STS client.

Regression Issue

• Select this option if this issue appears to be a regression.

SDK version number

3.911.0

Which JavaScript Runtime is this issue in?

Node.js

Details of the browser/Node.js/ReactNative version

v18.20.2

Reproduction Steps

1. Create the following content in ~/.aws/config file:

   [profile second]
   credential_process=/usr/bin/get-credentials
   region=us-west-2
   
   [profile first]
   source_profile=second
   role_arn=arn:aws:iam::123456789012:role/Role
   

2. Use the SDK with the first role: AWS_PROFILE=first npm run start (start invokes the following script)

   import { S3Client, ListBucketsCommand } from '@aws-sdk/client-s3';
   
   const s3Client = new S3Client({
       defaultUserAgentProvider: () => Promise.resolve(["my-user-agent", "1"]), // or customUserAgent
       logger: {
           ...console
       }
   });
   const command = new ListBucketsCommand({});
   const response = await s3Client.send(command);
   

Observed Behavior

Observe two AWS requests in CloudTrail. The AssumeRole request retains the default user-agent. The ListBuckets request contains the custom user agent:

1. aws-sdk-js/3.911.0 ua/2.1 os/darwin#24.6.0 lang/js md/nodejs#18.20.2 api/sts#3.911.0 m/E,w,v vs
2. my-user-agent m/E,T,i,o

Expected Behavior

Both requests contain the my-user-agent m/E,T,i,o user agent value.

Possible Solution

Propagate the custom user agent values to the nested STS client.

Note: other configuration to the outer client, like custom middleware, is not propagated to the nested client as well

Additional Information/Context

No response