I am attempting to recover an older (mid 2013) iMac

The computer stalls out about ¾ of the bootup. Booting into recovery mode and attempting to install the os from the internet failed because there isn't enough space on the drive and the drive itself in recovery mode is r/o. I created a MacOS Catalina installation on an external drive and booted the imac from it without problem. I then reformatted the imac harddrive after booting up from the external drive. I attempted to install the os using the external drive but without success. Booting the imac into recovery mode and attaching the external macos, attempting an installation that way failed because the drive didn't show up in /Volumes but is recognized by diskutility. Does anyone have info about restoring Catalina to the iMac? I can't use my macbook pro to restore since its running Monterey and won't boot into catalina from an external drive so that I can back it up to timemachine and restore the imac from timemachine. Please help thanks