I intended to Sort This Out.
And here's a caveat: I'm located in the UK, so the clock radio I'm using (Pure Siesta Mi) will probaby not be available where you live, but I reckon that pretty-all most manufacturers will use a similar system.
The first thing it to find out how the light or dark is detected.
Step 1: Finding the Light-detector
This clock was easy to open with just four screws in the bottom. Once they were out a small screwdriver was used to crack the case open. The insides are very neat, with one circuit board containing everything, a display, and a loudspeaker that just lifted out.
And there's the light-detector, in this case a photo-diode.
How do I know it's a photo-diode?
Well, the biggest reason is that it looks like one, but the tell-tell markings of 'D1' next to it on the circuit-board kind-of gave it away. The other reason is that it's a lot cheaper to use one of these beasties than it is to use a LDR (light-dependant resistor).
So, what is a photo-diode ...?
Step 2: Photo-diode Dimming, and How to Eliminate It
To check that this is how the circuit worked I covered the phoo-diode so the display dimmed, then shorted out its pins on the circuit-board with a screwdriver. And sure enough, the display came back to full brightness.
And so I put a big blob of solder between the pins! Job done, put the case back together, everything hunky-dory.
But .... be careful with other makes of clocks. I knew that the display on this one was being switched via the main processor because it switched the display between discrete levels of dimming, but other clocks may be not be so intelligent. Use a resistor to bridge the photo-diode if you're not certain.