Good thinking on the concrete, but how much would you need, and could you get it into the hole cleanly? You'll need between 2.5 and 3 cu. yds. of concrete to fill the hole, and a cu. yd. of concrete usually weighs over 3,000 pounds.
I lean towards the clay because you can incorporate it into the surrounding soil, knitting it together with the soil, and knitting each layer you add with the one beneath it. A liner placed on that area might allow water to get under it, negating any positive effects it may have. I was under the impression that you couldn't line just a small area, the liner had to come up above the water level around the whole pond.