I used to fish a lake very similar to the one you describe myself a few years back. It was an old estate lake where the commons are more dominant due to natural spawning. In fact, I have a water that I fish a lot in the summer that is similar again. It seems to be the theme with a lot of these estate lakes. Unfortunately, there isn’t an easy answer to this one. Or at least I have never found it!
However, I would still be looking to gain as much information about the habits of the few mirrors that are in the lake. Where have they been caught from in the past? Is there an area that they regularly show themselves on hot and sunny days? This might create an opportunity to stalk or surface fish for them which can be much more selective. When you think of the situation it is no different to a big fish angler targeting the biggest in the lake. Find out where the mirrors like to frequent and try to target them in these zones.
Another approach would be to go for the numbers game like you suggest. If information on capturers of the mirrors is at a premium then I would adopt this tactic. Start applying bait to areas and build up their confidence. Eventually the mirrors will turn up amongst the commons in a competitive feeding situation. Good luck, I know how difficult targeting them can be!