Failure to update to OS Monteray, insuffient memory, stuck in cycle

To put it plain and simple, a Triage mode doesn't exist. You could put your Mac into recovery mode and reinstall the OS, but that doesn't solve your needs, as in recovery you can't access your files to selectively decide what to keep and what to delete. Thus to reinstall the OS, since memory is full, you would need to wipe all data.

The alternative would be to start your Mac from an alternative medium, such as a pendrive or external hard disk, where there is a working operating system and from there access you internal drive as if it was a side storage.

You would need to have or buy a drive, boot in recovery mode, https://support.apple.com/en-us/HT201255 , format your external drive as Mac Os GUID partition and choose it as destination for installing a new operating system. Once that's done successfully, you can choose the external drive to boot from by pressing the Option key at startup and from there you will eventually be able to see and modify your files in the internal drive. Good luck, hope it helps.