What can I use for the PowerShell notin operator in v2.0?
You can use -NotContains in PowerShell v2.0 and up. The only thing you will need to do is just reverse your check. Using -NotIn you are comparing your current value to the collection, with -NotContains you do the reverse by comparing your collection to your current value.

So just change this:

Where-Object { $_.DatabaseName -notin ($NoBackups.DatabaseName) }


To 

Where-Object { $NoBackups.DatabaseName -NotContains $_.DatabaseName }


A rough example: