Hi!
Out of all of the ideas you suggested, I recommend the fan. It's likely to do the most good of anything, especially since there are probably already heatsinks in there.
The two things that will produce the most heat in these are the power supply and the video decoder/system processor IC. Capacitors in the power supply may be situated too close to hot-running components or of poor quality. If that's the case, I'd actually suspect some combination of those issues. Provided nothing blows up, bad caps could be replaced with better ones and that might be all you'd have to do.
On the other hand, if the video processing system on chip is getting too hot and burning out, that's going to be almost impossible to fix. Some run cooler than others...the LG chips seem to run so cool they don't even require a heatsink while the ones from Zoran must have a heatsink and it gets so hot that you can't touch it after just seconds of operation.
If you have a simple voltmeter and know how to use it, you should be able to find suitable power supply voltages to use for the fan. Be extremely careful--parts of the power supply are connected directly to the power line and will buzz/hurt/kill you. Voltages may also be marked on separate boards.
If yours has the power supply and television tuning circuitry all on one board (as some do), you'll have to trace the circuit to find out where the power supply outputs are at.
William