Running Bir Sur and would like to back up to external hard drive

Sorry, but you can't change the location of the backup for your iPhone.

But, if you manage to free up space on your HD, to create at least one current backup, you can copy that folder to your external HD and delete the backup afterwards in the settings/devices in iTunes.

In case you need the backup, just copy the folder backup into the same location your HD.

Check this article about it:

Locate backups of your iPhone, iPad, and iPod touch - Apple Support

What is “Other” storage on a Mac, and how can I clean it out? as a Start to make space

How to free up storage space on your Mac

• See used and available storage space on your Mac Especially the section Other: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.

And one needs to keep at least 15 GB of Empty Space Available at all times for the OS to load and function as intended. Not keep this space available may lead to eventually the computer will stop booting at all. Then that will present a different problem where Target Disk Mode will have to be encoded to clear some space just for the computer to boot.