How many flat tires do you get per year? Do you change them yourself or have GM change them under warranty? Why is it so important to not use a donut tire for a day or two until you replace the flat tire?
Email them a copy of the Service Bulletin and ask them to install the update on a USB drive, per the instructions. Then they can mail you the USB drive and you can install it yourself. Otherwise a dealer would have to do it and probably charge you around $100 for their time.
If you see CarPlay in the settings but not Android Auto, it means you need the software update. Android Auto will not work until it is updated. Normally you don't have to remove the hardware. It can be updated via the USB or through the OBD socket. See the attachment.
You should be able to upgrade the software without sending them the hardware. Either the Buick dealer can do it or you may be able to access the GM intranet for a fee.