You may have already answered this, but… have you renewed the heatsink’s thermal paste? (Ahh okay, you did.) Maybe try it again, unless you’re confident that it was done absolutely perfectly, without any doubt. Or, if you provide pics as mentioned here down below.
“With the system open and no active cooling on the CPU, the heatsink gets hot very fast, so I feel like the thermal interface is fine. It definitely does overheat in this state and the system crashes quickly without even needing to launch a game.” AND
"…left the machine open and placed a loud 120mm fan on top of the CPU, running at full speed and blowing straight upward".The idea of the supplemental fan was to blow down and or across the heatsink. If you had it positioned down on the heatsink and blowing UP-wards… well that’s entirely different and not like your original Sonnet configured, two-fan directional orientation either. I could be wrong in your case - but all fans here on CPU upgrades blow down or across the heatsinks.
“
…blowing directly on the heatsink”
“I think it does better with the case closed because the PSU fan is basically right above the heatsink…” Likely blowing down onto or across the heatsink, maybe?
As for jumpers, it might be helpful to have clear, top & bottom pics of the Sonnet. And if you do take these pics, a great time to check your thermal paste job for voids, etc.
after also removing the heatsink for the pics.
AND I absolutely
hate to recommend anyone spending any money in these situations, but there are fairly low-cost compact IR thermometers available. And in your case, when it crashes… you could open it up and quickly take the temp of the heatsink’s base, nearest the processor. Then, if it’s not terribly high, your assessment of it
NOT being a thermal issue may be correct.

And if you are correct, might be time to take the machine all the way back to basics and clean reinstall all of your software and the CPU upgrade supporting rituals. And of course, blow out any accumulated dust, especially in areas that you might not be able to clearly view without further disassembly.
There’s a picture around here of someone’s Quicksilver with a great ball of unseen hair amassed behind the motherboard.
AND maybe (only just maybe) time to replace the single system fan too? (Before OS re-install.)
Sorry, no magic bullets here. But we do try.
Hope that you get it all figured out.