You can write a PowerShell script that calls the Azure Management REST API and modifies the connection strings.
$restEndpoint = "https://management.core.windows.net/$subscriptionId/services/webspaces/$webspace/sites/$website/slotConfigNames"
$appSettings = $appSettingNames -replace ",", "','"
$connectionStrings = $connectionStringNames -replace ",", ""","""
$payload = "{""AppSettingNames"": [""$appSettingNames""], ""ConnectionStringNames"": [""$connectionStrings""]}"
Invoke-WebRequest -Uri $restEndpoint -Body $payload -CertificateThumbprint $certThumbprint -ContentType "application/json" -Headers @{ "x-ms-version" = "2014-04-01" } -Method POST
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