I ran into issues with this in the past, where Cachy would boot, but I couldn’t get into LM. As the Wiki suggests, osprober likely would have fixed that, but I had security concerns about using it.
For LM 22.1 and Cachy 6.16, installing Cachy (using Grub) first and then LM worked flawlessly with manual partitioning done according to the guide on linuxmint.com. Hope this informal tip helps another novice out there.
version Grub are different from LM and Archlinux side
you are using grub-customizer based on file cfg ,
this is not the case for grub vanilla.
You’re probably right. I selected grub as the bootloader when installing Cachy first, and LM uses some form of it at least. It may not matter which loader you pick for Cachy, as long as you have Mint install its boot loader over it. All I know is that method worked for me.