I've just checked the GRUB source code and I think it will clear the .bss section even if it's loading an ELF file.

grub-core/loader/multiboot_elfxx.c has a function named grub_multiboot_load_elf32/64 which actually loads the segments of the ELF file. A segment has two fields defining its size: filesz (which is the amount of bytes to copy from the file) and memsz (which is its actual size once loaded). If memsz is greater than filesz, it zeroes the trailing bytes:

  if (phdr(i)->p_filesz < phdr(i)->p_memsz)
    grub_memset ((grub_uint8_t *) source + phdr(i)->p_filesz, 0,
      phdr(i)->p_memsz - phdr(i)->p_filesz);

The .bss section is placed by the linker at the end of a segment and increases memsz by the size of it (but not filesz, to avoid having to place lots of pointless zeroes in the ELF file) - for example this is one of the segments from my kernel's ELF file, which contains the .bss section at the end:

  LOAD off    0x0000000000020000 vaddr 0xffffffff8011f000 paddr 0x000000000011f000 align 2**12
       filesz 0x0000000000004be0 memsz 0x0000000000017678 flags rw-

Here you can see memsz is 0x17678 bytes and filesz is smaller at 0x4be0 bytes. The difference between them is the size of the .bss section.

grub_multiboot_load_elf32/64 is called in the case when the address tag is not present, so the .bss section will be cleared by GRUB in this case as well.