Solved!

Although it is supposedly undocumented by Microsoft (or possibly a bug in Azure App Service), I added the following statement to my startup script:

cp /home/site/wwwroot/<path to your custom ini>/<custom ini>.ini /usr/local/etc/php/conf.d/extensions.ini

Setting the PHP_INI_SCAN_DIR environment variable was not enough to customize the PHP settings. Therefore, I had to manually copy the ini file to the PHP settings location.