A Business Rule cannot be launched manually, it can be triggered only by the event before/after which it is configured to be run. Also, a Business Rule cannot be configured to be launched by two different events simultaneously.
However, you can achieve what you want with the help of a Custom Command. You can create a new Custom Command and move all the actions and conditions of Business Rule  to the Custom Command.
Then, you can call your Custom Command both from the Business Rule triggered after user creation and from your PowerShell script.
To create a Business Rule that executes a Custom Command after user creation:
- Create a new Business Rule.
- On the 2nd step of the Create Business Rule wizard, select User and After Creating a User.
- On the 3rd step, add the Execute Custom Command action and click Select.
- Select the Custom Command you need.
To launch a Custom Command from a script, you need to use the ExecuteCustomCommand method exposed by the object for which you want to execute the Command. To execute a Custom Command on a target object of a Business Rule, Custom Command or Scheduled Task, you need to use the built-in $Context variable, for example:
$target = $Context.BindToObjectEx($Context.TargetObject.AdsPath, $true)
$customCommandId is the ID of a Custom Command. For information on how to get it, see the following SDK article: http://www.adaxes.com/sdk/?HowDoI.GetCu ... andID.html.