So I had a closer look tonight at that light bulb check module that I've got which came out of an old Audi...
Here's a view of the opened light bulb check module (VW/Audi part no. 4BO 919 421):

At its core seems to be a Philips Quadruple exclusive-NOR gate shunt resistor (HEF4077BT). I'm not sure what to make of the IC in the center, but maybe you guys know more.
Anyway, here's a look at the pin terminal numbers:

Pin terminal numbers on German cars are standardized based on the DIN 72552 industrial standardization directive. For example, you can see pins 56bL and 56bL1 on the connector. 56bL stands for low-beam headlights, left hand side.
This module has one 56bL and one 56bL1 pin, meaning one of them must be where power goes in from the car battery and steering column light switch, and the other one is where it comes out again, and on to to the light bulb itself.
This module gives out broken light bulb warnings on the KS and KB pins in the top right corner of the picture. But, and here's the convenient part -- accoding to all the research I have done on Audi electrical circuitry tonight, these aren't data lines. Instead, one pin is for all of the exterior lights around the car except the brake lights, and the other one is for the brake lights. If pin KB goes high, it means there's something wrong with the brake lights, and if KS goes high, one of your taillights or headlights is broken.
This essentially means that you've only got two kinds of warnings that you can produce on a CarDuino screen, and it doesn't enable you to tell which of one your many exterior light bulbs is broken. It also won't tell you which brake light is gone.
But, as I've said - this would give me a ready-made module, an officially approved piece of electrical engineering to put in my car. Not that I don't have faith in my own ability to build and solder together my very own homemade light bulb check unit that can do the job, but using the Audi module would be an extra bit of reassurance.
And besides, here in Germany, these modules can be had used for €5 a piece on eBay. It'd be a challenge building my very own light bulb check module for that kind of price, taking into account all the different parts I'd have to buy...
The connector plugs could be slightly more difficult to come by, I will probably have to go to a few junkyards here in the area, looking for late-90s Audis where I could just cut them out of the central electrical panel looms.
EDIT:
I just thought of another approach... what if I pass a small 5V current from the Attiny (!) through the optocoupler, using a 10K resistor or similar, and then connect the GND side of the optocoupler to the wire going to the light bulb? That way, the optocoupler would be grounded through the light bulb, and would stay LOW if the light bulb is broken.
One issue that I am imagining with this approach though would be that it would affect the light bulbs' lifespans if there's a current constantly going through them, but again, that current would be minimal.