I noticed that when I transmit from my handheld Baofeng Radio, with the standard rubber duck antenna, that my laptop screen goes blank.
This only occurs if I'm within a few feet, and even on powers down to one watt.
As an electrical engineer: that's still a lot of power if just wanted to communicate over a couple meters, but:
If your laptop has any certification (it has, unless you illegally imported it), it should be tolerant to any let's say "non-malicious" interference.
So although I've played around with the cheapest Baofengs you can get on amazon.de and saw how terrible the spurs were, I'm pretty sure this should not happen.
My question is if there are standard steps that can be taken to attempt to diagnose what's going on here, and if so, how can I do them without buying a bunch of extra equipment?
Well, you've diagnosed the problem: The laptop breaks its FCC specification by being susceptible to lawful interference.
Now, if I were in any department of ASUS, I'd be mightily relaxed. As laptop manufacturer, I'd deal with a couple hundred upset customers per day. The least of them have tried to irradiate my devices with so much RF power that things start to fail, and it's (albeit not legally sound, probably, but I'm really not a lawyer) really easy to argue that you just shouldn't operate a strong radio next to electronic equipment.
Therefore, I doubt there will be an engineer sitting down with your laptop model trying to reproduce the issue and nail down the source. However, if he did, he'd probably use exactly your radio first.
The problem here is that you forgot to mention on which frequencies you operate. Radio modelling becomes easy when you're in the far field, ie. radio waves propagate more or less as plane waves, or at least as surfaces of a growing sphere around the transmitter. In the near field, i.e. for distances between transmitting and receiving device below or in the order of single wavelengths, things get a lot more complicated.
So, next thing that engineer would do, if this is far-field, would be record the emissions of your radio (because, reproducibility) and analyse them. He would then either use your radio or some software defined radio (signal generator) to reproduce the waves, and feed them into a highly directional antenna and try to figure out what to point at to cause the outage.
I'd however presume this is near-field. That would make it a lot less worse – because usually, devices don't need to be "near-field secure". No-one demands it's safe to operate your laptop right next to a 1 kW HF station :)
So, in that case, I'd take a signal like generated from your radio, use a lot less power, and feed it into a simple dipole (trying to reproduce the mismatch/matching of your rubber ducky antenna and the Baofeng, but measuring that is another task) that I attached to a plastic/wooden arm, to minimize effects of my own body. By observing the SWVR of while I hover the dipole over the device under test, I could find the regions where power couples out of my dipole into parts of the laptop most easily. Then I'd really look at these – it's not too bad if that were e.g. the metal hinges on the laptop screen, and I'd just tell the R&D engineers to take care about that in the ground design of the next laptop.
If, however, something like the very sensitive display signal lines running through the edge of the display absorb relevant energy, and the energy doesn't seem to be absorbed by a ground plane below those, that would actually be a reason to talk to the engineer responsible for that design – for signal lines, proper ground planes are crucial for reliable operation, and the support cases where this, out of sheer bad luck, happens to someone in a situation where he's not holding the radio himself, will cost my company real money.
this laptop basically falling apart anyway,
ah, so that is a good hint: Check that all the case insides are properly connected to each other. Laptop cases around critical electronic parts are often metallized on the inside and often have small copper springs or sponges with a copper mesh around to make contact when you properly close the case, thus forming an effective shielding. If your laptop lost its mechanical integrity, this effect might be compromised.
The same goes for signal connectors, by the way, and in your case, especially the LVDS connector that goes between mainboard or graphics card and your screen. Digital signal receivers are pretty good these days and might actually work OK with a "half-connected" LVDS pair, but if significant interference happens, that might kill the decodability. So check your connections!