I was thinking if you go the concrete route and hit the leak, there may be enough flow and pressure that the concrete won't set well as it will have too much water pushing into the mixture. You will probably need to get some plastic water barrier down there to prevent water penetration until it sets up. After that, the plastic can degrade, who cares. So any big old roll of plastic like that from the hardware store will probably work to bag it up. However I am uncertain of success since there will be a gap between the plastic and the soil.
One thing about digging the trench, if there is a good leak, you will probably see it while digging, and be fairly certain of making positive progress.
I am having a head-scratcher of a leak problem myself, and would love to get the the bottom of the issue to get it fixed (similar to yours). All was fine and dandy with plenty of inflow, but now that someone turned off the rain faucet, I am seeing some pretty significant drops in level vs. time.
