There are two things that I have picked up over the years of puzzle solving that eventually made it possible for me to solve this one. One is EPSx3. EPS means Edge Piece Series. It was so named by Philip Marshall in his Ultimate Solution for solving the Rubik's Cube. EPS is a simple 4 move sequence that cycles 3 edges. It also does a double swap of corners. EPSx3 then, would cycle the 3 edges 3 times, which puts them back to solved, if they started solved. But it does the corner swaps 3 times also, which is like doing it once. That means you can move corners around using 2 adjacent faces. :D

FRUR'U'F' was a simple way to orient the yellow edges, and Sune (R U R' U R U2 R') was used to move the yellow edges into place. Long ago I discovered that doing Sune right-handed then left-handed could be used to twist two corners without moving edges. I'm not sure why I don't use it more, but of the three ways I know of to twist corners, I never use this one. But the other two have to have parallel faces that can twist, and on the Fuse Cube, that is not an option.

During the first solve I somehow managed to go from needing a corner 3-cycle to a double swap. Now if I could only figure out how I did that, I might be able to solve it again...

Oh yeah! Swap two of the 3 that need to 3-cycle and 2 others that won't solve either one. Using this technique I have solved the Fuse Cube several times.