The quick and dirty way to fix the fallback thing is likely to be an incorrect entry by the grub os-prober from the other system. If you change the line:
initrd /boot/amd-ucode.img
(obviously if Intel powered, it will read /boot/intel-ucode.img) to
initrd /boot/amd-ucode.img /boot/initramfs-linux.img
in the file /boot/grub.cfg it will probably ‘fix’ the problem.
The reason this is ‘dirty’ is that the file WILL be overwritten on every grub update, and need to be ‘fixed’ again.
What I would do myself is install rEFInd, and let it do the whole thing. We have a wiki entry on that right here, and nothing is required beyond the install to see if it works.