Okay. I figured out what needs to be done to get it to work.
First, I needed to change the account for the SolarWinds Alerting Engine service from Local System to the AD service account that I use with Orion.
Second, I need to bring up a 64 bit command prompt as Administrator. Browse to C:\windows and run the command mklink powershell64.exe c:\windows\system32\windowspowershell\v1.0\powershell.exe.
Once I did this, if I use powershell64.exe -command "Path of powershell script" in the Execute an external program, then the script loads the add-in successfully.
The problem is that the Advanced Alerts is a 32bit process. So even if you point to the path of a 64bit program, Windows runs it as a 32-bit process. But if you use a symbolic link, you can invoke the 64bit process.