Hi,
I am not very good at the old command line, if the solution can be all in PowerShell, then I might be able to help.
First of all, I am not able to test this in a real environment as I do not have the SSAS installed in my computer and I have never used the Invoke-ASCmd cmdlet, so I will be using the cmdlet as you typed in your post - assuming it is correct, although instead of using the parameter -InputFile, I will use the parameter -Query, based on the description in HERE.
$DBNAME = "ANALYSIS_DB";
$SCRIPTPATH = "E:\cube_backup.xmla";
$SRC = "ServerName\ServerInstance";
$BACKUPPATH = "G:\Cube_BackupFolder";
$date = Get-Date -Format "yyyyMMdd_hhmmtt";
$BACKUPFILE = "$BACKUPPATH\$($DBNAME)_cube_$date.abf";
$query = @"
<Backup xmlns="http://schemas.microsoft.com/analysisservices/2003/engine">
<Object><DatabaseID>$DBNAME</DatabaseID></Object><File>$BACKUPFILE</File>
<AllowOverwrite>false</AllowOverwrite>
</Backup>
"@;
Write-Host "Invoke-ASCmd -Query `"$query`" -Server `"$SRC`"";
Invoke-ASCmd -Query $query -Server $SRC