Anyhow, moving on.
Anticipating heat being a concern in such a tiny box, I'd picked up a 120mm 920rpm fan as a case fan: The large diameter and low RPM allow a substantial amount of air movement (optimally 45CFM) to push more air through. Mounted in the second hard drive bay on the side of the chassis, it moved a fair amount of air through the case, but alas, heat was still an issue.
Once running, the CPU would steadily climb up to around 45 degrees C at idle, with the CPU cooler fan pushing up to maximum speed (which, incidentally, sounds remarkably like a jet engine). Under load, the system would overheat and shut down. I tried dropping the clock speed down to 1800mhz (the A6-3500 runs at 2100mhz stock, with a "turbo" speed of 2400mhz for one core) and the voltage down to 1.3v from 1.4. Unfortunately, while it did run a bit cooler, still not sufficiently.
Obviously, this was untenable. Having your HTPC shut down in mid-movie isn't exactly the behaviour you want from your HTPC, unless you're watching something spectacularly terrible, and having to turn up your TV to hear it over the miniature jet engine wailing beside it detracts from the experience.
So, I pulled out the power supply. The ridiculously long cables then became an advantage, instead of a disadvantage: I could run the power supply entirely out of the case without any trouble at all. Now, I'm not considering staying this way, but I wanted to be sure that with the power supply out of the picture, the system would work out well.
And, fortunately, it does. With the power supply out, the CPU fan at 40% speed (whisper quiet), the system runs at 36c under load. Experimenting, I was able to overclock the system clock to 2500mhz (on all three cores, just disabling the silly "Turbo" thing), the GPU portion of the processor from 450mhz to 550mhz, and the RAM from 1333mhz to 1600mhz, with core temperatures not exceeding 39c under load.
So, moving forward, I'm replacing the power supply with a 120w picoPSU from Short Circuit, basically just a plug in your box and a laptop power brick. 120w is much more than is needed, with the full system drawing at most 85w under stress testing (~60w during playback) but the next one down, 90w, is uncomfortably close to the peak power draw from the system.