There is no way in the UI or via an out of the box PowerShell command to change the application pool used by a web application. Don't bother trying to create one manually in IIS and setting the web app to use this, it won't work because the application pool does not exist as a known object in the configuration database and you'll get 500 Internal errors when you try to open the site in the browser.
You'll first need to create a new application pool using the SharePoint object model (which in turn creates the app pool in IIS) using either PowerShell or compiled code.
Compiled code example: http://code.msdn.microsoft.com/Create-a-new-application-e951983d
then you can change the web app to use the new app pool...
PowerShell example: http://gallery.technet.microsoft.com/scriptcenter/Change-SharePoint-Web-64b872d7
Finally you can delete any unrequired application pools:
PowerShell example: http://gallery.technet.microsoft.com/scriptcenter/Delete-a-SP-WebApplication-9a7d82e4
Your first statement is not entirely true "There is no way in the UI or via an out of the box PowerShell command to change the application pool used by a web application.".
You can use PowerShell to to change application pool of an existing web application.