I resolved the issue by changing the storage account's network from selected networks
(in which I'd whitelisted by ASE's VNet and the outbound IPs for the web app) to all networks
via the portal:
https://portal.azure.com/#@<myTentantName>.onmicrosoft.com/resource/subscriptions/<mySubscrption>/resourceGroups/<myStorageAccountsResourceGroup>/providers/Microsoft.Storage/storageAccounts/<myStorageAccountName>/networking
I then restarted my app (including dropping and re-adding the mount... not sure if that was required, but for good measure):
# stop the web app
az webapp stop \
--subscription <sub id> \
-g <resourcegroup> \
-n <sitename>
# drop the existing path mapping
az webapp config storage-account delete \
--subscription <sub id> \
-g <resourcegroup> \
-n <sitename> \
--custom-id <path-mapping-name> \
# re-add the path mapping
az webapp config storage-account add \
--subscription <sub id> \
-g <resourcegroup> \
-n <sitename> \
--custom-id <path-mapping-name> \
--storage-type AzureFiles \
--account-name [Azure storage account name] \
--share-name [Azure storage share name] \
--access-key [storage access key] \
--mount-path [/path/to/mount within the container]
# restart the web app
az webapp start \
--subscription <sub id> \
-g <resourcegroup> \
-n <sitename>