I meant the following:
- Find out the Debian package is too old
- Create Arch Live USB
- Boot Arch Live USB
- Copy GRUB config from the Debian install to the current Arch live system
- Install the up-to-date GRUB while in the Arch environment
The bootloader installer package is distro dependent, the bootloader the package installs isn't. You can boot Debian no matter if the GRUB is installed from Debian stable, Debian Sid, Arch, Fedora or even FreeBSD. Otherwise, dual booting wouldn't work.
Like I said, I've done that before, though with SystemD Boot instead of GRUB, which was a bit simpler due to how the bootloader is configured.