With two installs you have to normally do two updates. Whichever system's grub is in MBR must be booted and sudo update-grub
run to find newer kernel in second install.
But since you have two drives, put main working install's grub into the MBR of sdb. And keep test install's grub in MBR of sda. When you install grub defaults to sda. Better to use Something Else and on partitioning screen change to MBR of same drive.
You can boot into install in sdb and install grub to the SSD's MBR. Then change BIOS to boot SSD.
sudo grub-install /dev/sdb
sudo update-grub
If you can only boot from sda, then just change /dev/sda.
But if you originally installed grub to sda, it remembers that setting for reinstall on major updates. So you also need to reset that:
To see what drive grub2 uses see this line (BIOS only, not UEFI) - grub-pc/install_devices:
sudo debconf-show grub-pc
It will show drive model & serial number
to see drive info
sudo lshw -C Disk -short
to get grub2 to remember where to reinstall on updates:
sudo dpkg-reconfigure grub-pc
Enter thru first pages,spacebar to choose/unchoose drive, enter to accept, do not choose partitions
You can also avoid the double update by manually adding your own boot stanza in 40_custom and directly boot partition or link to most current kernel.
https://help.ubuntu.com/community/MaintenanceFreeCustomGrub2Screen
UEFI systems do not use grub-pc but grub-efi-amd64 if 64 bit or perhaps signed versions. So all of above is only for BIOS based systems.