In my case I would power on the PicoPSU with a paperclip shortcircuiting two pins in the 20/24 pin connection cable.
How do I control the PicoPSU only using the 4 pin power -> 12v Molex cable?
What you could do if you want to use the PicoPSU or another ATX power supply is to hook up to the ATX12V connector and then attach a switch between the green wire on the 24-pin ATX connector and one of the black connectors. The PicoPSU does not have the colours so this may not help but basically by shorting these pins the PSU will start up. This is how the motherboard starts up the PSU in computer normally.
I would also suggest you don't use the PicoPSU because you don't need it. Have you asked the SBC OEM for PSU suggestions? There are many 12V open frame PSUs available from component specicalists like RS.