I would set up the new disk with an EFI partition. It wont take up much space, and gives you the ability to easily move the disk without having to change existing parition sizes to add an ESP later. I would use the --removable on the grub-install to put a copy of the grubx64.efi bootloader into /EFI/Boot/bootx64.efi. (If you are using secure boot, you theoritically should use --uefi-secure-boot, but last time I tried it, that did nothing, instead of putting shimx64.efi in /EFI/Boot/bootx64.efi and a copy of grubx64.efi. So you'll still have to manually copy in the correct files).
Your other potential problem is the disk caddy. Grub may have problems when the caddy is present -- like freezing for a minute, or maybe indefinitely. This may occur even if the caddy is not used for boot. A workaround is to have a device like USB you can put earlier in the boot order than the hard disks, but you might still not be able to reference the caddy until you boot, so good luck. The caddy will still be usable as a storage device.
See the comment about moving the files. Nothing special for an
SDD in a caddy.