Is There a Way to Send Texts from an Alternative Extension/Phone Number?

Yes, you can add more direct numbers to an existing user extension. Login to your account portal with an admin user's credentials, select a user and see the screenshot below to add a direct number.

No, you cannot send SMS on behalf of another person within a user group and using the RC app. However, you can implement your own app using the /sms or /batches (for high volume SMS) API to send SMS messages. Your app can be a password flow authentication which always uses the user name and password (or JWT token) of the user who owns the SMS phone number for sending and receiving SMS messages. Now, you can control your app's SMS feature access by requiring users to login the app with their own credentials, then detect their identity (e.g. extension id) to allow the user continue using the app or not.

Think about a fax machine in a locked room and you allow only certain employees to enter the room to send faxes.