A follow-up to this issue.
I've continued to have intermittent issues with the SD Mini going blank for days at a time, then coming back. All this time, the buttons function; they just don't show anything. Meanwhile, the SD Neo works just fine.
However, with the update to the latest BTT 5.294, both SD Mini and SD Neo went blank. I tried the latest alpha update BTT 5.296 but this made no difference.
I rolled back to BTT 5.179 and the SD Neo remained blank for about an hour, then came back to life. I have not tried the SD Mini yet.
I have found that this process above of upgrading, then to alpha, then downgrading, has reverted some of my BTT settings to something from many months ago. I don't know why, but the changes in configurations that I've made from the last month or so seem have been lost.
Perhaps this process above might help you identify what is causing this. In particular, it was interesting that downgrading the BTT back to 5.179 didn't make the SD Neo come back to life immediately, but took about an hour to start showing the buttons.