I am seeing a similar behaviour, strangely.
When I connect BelaMini to USB battery the board boots fine.
If I connect BelaMini to a programmable power supply, supplying 5V (2A available) to pins P1.01 and GND (next to 5V on P1), I get a flickering of the power LED for a few seconds and then nothing.
If I connect BelaMini to 5V supply from a linear reg (LT1085) which is regulating down from 9V (within spec AFAIK) I get the same behaviour; flickering of power LED for a few seconds and then nothing.
I am making a guitar effects box, so I need to power so external JFET opamps, hence the external regulator and 9V drop down.
Anyone else had a similar problem?
I am certain this bench supply can shift the appropriate current, so there must be something else at play here.
Further, connecting a normal pocket beagle to the bench PSU using the same configuration confirms that this process should be working. I am assuming that I have damaged the OSD335x power input circuitry on my Bela mini, and that I will have to replace it when I can eventually afford it.