The kinetic energy doesn't disappear just because the neutron interacts with a nucleus. Any more than it disappears when a bullet hits a block of clay: the bullet mostly stops, but the clay moves and heats, because momentum is conserved. In the same way, ITERs breeding blanket is going to heat up plenty. Run water pipes through it and you're good.

The beryllium supply is a real issue though. CFS is working on a tokamak a tenth the size that should do the same thing as ITER, because it uses much better superconductors, but even that uses a lot of beryllium.