We have installed many successful basements in similar conditions.
Firstly, your footing should bear on the top of the clay, as long as soils tech approves the bearing capacity.
Notice that the water travels "on top" of the clay layer.
Excavating into the clay will create a "bowl" effect that will always draw water to it, and have you pumping to infinity.
From what you describe, you may have 3 ft of foundation wall showing, when done. Better that, than 6 in of water in your basement. LOL
Pour a 10 in thick footing and install a perimeter weeper tile system with clear stone coverage and interior sump box.
Place clear stone also,level with the top of your footings to create a water retention area under your basement floor. Make sure a drain tile is installed into this layer, and connected to the sump box.
We use a Stone Slinger to accomplish this. 80 tons of stone or better is not uncommon for an 1800 sq ft home. A Telebelt works fine too.
This clear stone will allow for many thousand's of gallons of water to be collected and pumped in season.
Use filter cloth on your tile, geo fabric over also on exterior tile . Air gap your buried ICF foundation exterior.
Vapor barrier underside of your basement floor, and better yet, install rigid foamboard underfloor also,to reduce heat loss thru the basement floor.
Payback is immediate.
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