I needed this and couldn't find a solution online so I sat down and figured this out last night.
In my use case I'm using this in a schedule to monthly get the newest QuickSupport on my Domain-Controller and from there on I distribute it to all domain computers' public desktop.
Note: Make sure to replace "<yourcustomurl>" and the destination path where it should be saved to.
$iebrowser = New-Object -ComObject InternetExplorer.Application
$iebrowser.Navigate('https://get.teamviewer.com/<yourcustomurl>')
Start-Sleep -Seconds 3
$CustomTVQS_URL = $iebrowser.Document.getElementById('MasterBodyContent_btnRetry').href
Start-BitsTransfer -source $CustomTVQS_URL -Destination C:\test\TeamViewerQS.exe
PS: I put a 3 second pause after calling the website URL since it takes a little time to create the ComObject and call the website URL. Powershell is working through the commands too fast and otherwise throws an error that the $CustomTVQS_URL is a null object. You may increase or decrease this if needed.