I doubt flashing an earlier version of the BIOS would do any good. Is the install from that CD/DVD that you are using "right now" installed on a newer machine? And if so, is it installed in Legacy BIOS mode or UEFI mode?
The Fedora install media iso's are created so that they are compatible with systems running in either Legacy MBR mode or in UEFI mode. On a system booting that install media (either USB stick or CD/DVD) that is booted in UEFI mode, the media will appear twice in the UEFI's device boot menu, once for non-UEFI and once for UEFI. I think the first of the two entries is always for non-UEFI, so if your HP Probook is truly non-UEFI it would automatically attempt to boot the first entry.
Secure boot also does nothing to prevent a storage device from booting. What it does, when enabled, is to prevent an unsigned OS, kernel or kernel module from loading into the system. Since you've successfully installed from that CD/DVD media on another machine, that should eliminate the media itself as the culprit, but not the CD/DVD device itself unless you have already used the CD/DVD device to successfully boot other bootable media and/or access data CD/DVD's without problem.
Wish I could offer some more definitive advice, but I can't think of anything more to try. Except to perhaps create a USB stick with the install media on it and try from that. The command line dd method gives the best results for that to transfer the iso image to the USB stick. If you have one laying around you can spare, give that a try. All data on the USB stick will be vaporized in the process.
# dd if=<downloaded_install_iso_file> of=/dev/sd<x> bs=8M
Where <x> is the usb device (unmounted). "if" is input file, "of" is output file, or device.