I am using AMI BIOS 2.16.1240 (dated 2015) on an ASUS H97-PLUS motherboard. In the BIOS, I go to Advanced > PCH Storage Configuration > SATA (SATA 5,6) and M.2 Configuration.
It offers a choice among Auto, SATA, or M.2. I choose SATA > F10 > Enter. On reboot, the system returns to BIOS. It appears not to find a bootable operating system. If I change that choice from SATA to Auto or M.2, the system reboots into the Windows 10 installation on the M.2 drive.
With that item set to SATA, I boot a live Ubuntu USB drive. It boots by default; I don't have to select it in BIOS. In Ubuntu, I open the Disks program and look at the drives listed. It does not list the M.2 drive. I reboot, change SATA back to M.2, select the Ubuntu USB, and boot into Ubuntu again. Now Disks does list the M.2 drive.
This PCH Storage Configuration setting appears to offer the kind of solution the OP is looking for. Does this option not exist in other BIOSes - or did AMI remove it sometime after 2015 - or is it just hard to find?