I see you have a backup of your RAID 1 volume.
What you can do is to access the RAID BIOS by pressing Ctrl+I during POST and then reset all disks to non-RAID. After that you can create the RAID 5 volume from the RAID BIOS. As soon as you create the RAID 5 volume all the information that remains in the hard drives will be completely deleted. Once you have the RAID 5 volume created you can install Windows and then Intel RST.
In case RAID 5 option is not available in the RAID BIOS then there is a chance an option called Intel Rapid Recovery is enabled in the BIOS. When that option is enabled a RAID 5 volume cannot be created.
All disks are currently set to non-RAID. When I go into the RAID BIOS, I can create a new array, but only RAID 0 or 1. I don't see any option of Intel Rapid Recovery (or similar sounding thing) in either the RAID BIOS or the system BIOS. Is there a specific place I should be looking? Or is it somehow possible that there's some other aspect or feature that is disabled on my particular system?
Another troubleshooting step you can perform is updating the BIOS of the motherboard. You can find BIOS updates on this link http://www8.hp.com/us/en/support-drivers.html.
If the issue persists I recommend contacting HP in order to check if they have disabled the option to create RAID 5 volumes in that specific computer model.
If you do not find any option in the BIOS related to Intel Rapid Recovery then there is a chance HP has hidden that option so; nobody can change it.