Ok ... that was a worthwhile exercise. The 3-in-1 diagnostic utility creates a bootable HP_TOOLS. This is exactly what I was looking for - hopefully it will work as well on your machine.
HP PC Hardware Diagnostics
Scroll down to : HP PC Hardware Diagnostics 3-in-1 USB Key
A few notes:
Don't sweat the Authorized service partners / IT professionals verbiage. This isn't something everyone needs, but you can certainly use it.
If you have any questions, ask ... the utility is pretty much the same thing as the built in pre-boot diags menu. I might have to refresh my memory, but at least it isn't completely alien to me.
1) It's a 2 GB download - took over an hour for me to get it.
2) The install first extracts the softpaq to a new folder, so you need at least 4 GB free space on a drive (download + extract)
3) Setup is in a command prompt and isn't very smart.
You have to tell it the drive letter of the USB. It doesn't offer any - so be sure you know the letter of the thumb drive.
4) It wasn't clear if it wanted the letter only or the letter and the colon. I'm not sure if it matters, but I entered the letter without the colon (it might figure it out if you include the colon, I just don't know)
!! WARNING !!
The drive letter you enter will be formatted!
Triple check before hitting enter.
5) Setup takes a while - 20-30 minutes.
If successful, you have to hit the any key to finish.
6) Read the documentation to become familiar with the utility
Here's a copy from the thumb drive I created.
HP PC Diagnostics USB Key User Guide - Rev 4 (Oct 2014 Release) - English.docx
7) When setup is done, you can boot from the thumb drive
Restart your machine
select f2 - run diags
The CSO and Agent numbers are just identification you make up.
I used the name of my computer for the CSO and Agent99 for the agent number
You can update the BIOS with this utility
... now all I have to do is figure out where it wants the data to do the update.
On my system with a HP_TOOLS partition already setup, the folder tree looks like this:
I expect the thumb drive will be the same and that the BIOS update would go in the New folder.
I'll get back on that. Let me know if you were able to bring up the diags using the 3-in-1 utility thumb drive.
After that, it's going to be a breeze. I should mention that this will bring your system up-to-date, and I think this will help ... but I'm uncertain if this will correct the issue. It seems like a BIOS update is the only thing that might - everything else looks in order.