I experienced the same issue and had to use the same work around on T-mobile. Once I purchased the app on my PC (even though it was free), I was able to download ok and apps would even show up automatically in the New/Install page in the Amazon Appstore on my android phone. Prior to my first successful download, I was only receiving a generic download error. After my first download, error now reads "there was an error purchasing XYZApplication. Please <click here> to verify your payment method (this will open your browser) and try again."
From my android phone, following the <click here> link in the error message took me to
https://www.amazon.com/gp/aw/vsd.html which also had an alert to "Please enable cookies in your browser settings to continue shopping" which I ended up ignoring. Signing in at this page then took me to 1-Click Settings page
https://www.amazon.com/gp/flex/aw/sis.html where it was noted Mobile 1-Click ordering is turned off. Entering a mobile number here AND selecting a payment method fixed the problem. Note the above changes were required even though the app I used to test this was free.
Sure would have been nice if the initial generic download error had been more descriptive.