Unable to update BIOS for XPS 13 9370 developer edition

Are you following the instructions below from the BIOS page?

Updating the BIOS from BIOS Boot Menu (independent of operating system)

Note 1: Before updating the BIOS, ensure that you suspend BitLocker encryption on a BitLocker-enabled system. If BitLocker is not enabled on your system, you can ignore this step. For information about how to disable BitLocker, see How to Enable or Disable BitLocker with TPM in Windows at support.dell.com.
Note 2: Do not turn off the power or interrupt the BIOS update process during the update.
Note 3: Your system requires a restart after installing the BIOS. The restart can be deferred but must be completed to ensure that the update is installed.

Installation
1. Copy the downloaded file to a USB drive. You do not need a bootable USB drive.
2. Insert the USB drive into any USB port.
3. Power on the system.
4. At the DELL logo screen, press F12 to access the one-time boot menu.
5. Select BIOS Flash Update in the Other Options section.
6. Click the ... button to browse the USB drive to locate the downloaded file.
7. Select the file and click OK.
8. Verify the existing system BIOS information and the BIOS update information.
9. Click Begin Flash Update.
10. Review the Warning message and click Yes to proceed with the update.
The system restarts and displays a Flash Progress bar at the Dell logo screen. The system restarts again when the Flash update is complete.

Yes, as evidenced by the screenshot I attached.

Obviously there are 3 invalid entry, hence the greyed out and the firmware update is not going to flash.  You probably know what change you made to the system.  You need to reset your system back to factory settings if you want to use retail BIOS in deployment mode.

I did notice these entries, however they're in the new firmware section, so I wasnt sure what to make of them.

Either way, I tried your suggestion, but the outcome remains the same.

To be clear I don't have anything modified on the system. I run Linux but that's it.

You can try again, download this BIOS and copy it to a USB formatted with FAT32 and follow the steps outlined by @JOcean to update BIOS using F12, it will be performed outside of OS. 

You are pressing F2 and the directions clearly state F12, and yes it makes a difference.

That was a typo, I'm using F12. F2 leads to the BIOS Setup page which (on my 9370) has no flash update  functionality.

Some users could not receive correct resolution because information were neglected or left out.  From your initial post, I could tell that the BIOS weren't update for more than 2 years already.  Beside addressing CVE, a BIOS update can help to resolve many performance issues but sometimes it also can creating problem.  If your machine has no flash functionality, there is no need to update BIOS.

Thanks, it helped me