That keyboard could be polling and causing the wake issues.
Unpair the A1314 and test using only a USB keyboard.
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If that works use a USB keyboard to login and reconnect the A1314.
Or... replace the old AA battery A1314 with a new Apple rechargeable keyboard.
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Plus full Computer Sleep Mode is known to cause problems.
As a test:
Turn On "Prevent automatic sleeping when display is off"

Set "Turn display off when inactive" to 5 or 10 minutes

Leave the Mac Mini On and only allow the display(s) to turn Off
Do not manually Sleep from the Apple Drop Down menu
Only Shut Down when away for a day or more