It’s no problem for rEFInd to use images in /efi, but it will not automatically generate a bootable entry for you until you run refind-install from within the EOS installation while the EFI partition is mounted. You need to do this step even though rEFInd is already installed on the EFI partition from your other system.
It will automatically recognize rEFInd is already installed and will preserve all of your settings, etc–no need to worry about messing up your other system’s setup. It should just add a bootable entry on the rEFInd menu for booting directly into EOS, and also for booting to systemd-boot (rEFInd can hand off the boot process to systemd-boot if you want to do that).