We built a 25 acre lake in 96 that is 25 ft deep. While digging the core trench we made sure that we dug down into solid clay for at least 3 ft. In the original channel area, there was a layer of yellow clay, but beneath it was a rock/rubble vein that was about 20 ft wide. My pics look a lot like yours, with water and the sidewalls collapsing in about as fast as the excavator could pull it out. We eventually dug through this underground stream and got into good clay.
We had 2 scrapers bring in clay, and a dozer was forcing the clay down into the base of the trench as the excavator was pulling out the junk. We actually excavated out quite a bit of the clay we were putting into the hole, just to make sure it was all good material. The contractor worked his butt off, but the pond is sealed and has not leaked (knock on wood).
If you can find good clay beneath, your best bet would just to try to complete the core as originally designed. I see another response in which they used tubes to divert this water, but it sounded like that was quite a project.
Good luck.
