The issue that you are reporting does not seem to be processor related since the processor either works or it does not work.
On this case, since even installing the latest BIOS version for your motherboard does not fix the issue, then I would suggest contacting your laptop manufacturer so they can assist you or provide any warranty option.
The laptop is out of warranty, apparently it costs like $69 to get them to figure out the problem.
Could this be something to do with 'Errata' that occurs in some CPU's?
I do not think that it is an errata issue, unfortunately the only way to determine the defective component is by taking the laptop to the manufacturer so they can perform the required tests.