Extracting data from iPhone backup on iCloud
If you don't want to use a third-party tool for an iCloud download, the only canonical Apple way is through a reset and restore.

As you say, it'd interrupt your workflow, but depending on the size of the backups involved you might be able to get it done in an hour. The greatest consequence of doing this is that you'd have to reauthenticate most apps on your device after restoring your backup.

Using iTunes backups as part of the switcheroo would make this as fast as possible, and you'd do something like this:


Use iTunes to back up your iPhone. (An iTunes backup will be fastest. Make sure it's an encrypted backup, as the unencrypted ones don't include everything.)
Back up your encrypted backup, or move it temporarily onto your desktop. It'll be in one of these folders.
Erase your iPhone. Settings > General > Reset > Erase All Content and Settings.
Restore the iCloud backup that you want to examine to it.
Back up the newly restored phone with iTunes. (This will overwrite your earlier iTunes backup, as it's the same device. That's why it was important to back it up earlier.)
Move out the restored phone's iTunes backup, and switch your original iTunes backup into the backup folder.
Erase and restore your phone from the original iTunes backup so that you're back to where you started. Log back into your assorted apps on the phone.


Step 5 will have created an iTunes backup of the iCloud backup data you want to examine. You'll then be able to examine that with a bit of hacking around, or use of a third party tool. Good luck. As someone who built the first iPhone backup tool (and later, iCloud backup tool), I'd say it's important to choose the tool you use carefully!