For anyone else looking to do this, I eventually used an Azure Function App on a timer, which runs Powershell commands to delete anything older than 168 hours (seven days). The basics of setting this up are here:
https://learn.microsoft.com/en-us/azure/azure-functions/functions-create-scheduled-function
I then used the following Powershell script:
param($TimerBlob)
$CleanupTime = [DateTime]::UtcNow.AddHours(-168)
$context = New-AzStorageContext -StorageAccountName <MyStorageAccount> -StorageAccountKey <MyAccessKey>
Get-AzStorageBlob -Container "<MyContainerName>" -Context $context |
Where-Object { $_.LastModified.UtcDateTime -lt $CleanupTime -and $_.BlobType -eq "PageBlob" -and $_.Name -like "*.bak"} |
Remove-AzStorageBlob
$TimerBlob is the name of the associated timer object.
It's been running for a couple of weeks now and seems to be working well. I hope this helps someone out there. :)