Note: use the arrow keys to manipulate the fixed level to the bottom side. Use U or U' to align a corner relative to the bottomlayer. Then use arrowkeys again to rotate the entire cube according to below situations
First move the corners into the correct position (not necessarily correctly oriented):
If you have 2 correctly positioned corners, then choose an incorrect corner to the front/right (and the correct ones to the back) until you get 3 incorrect (or 4 correct) corners after the move above.
Then orient the corners by one of the below situations:
Note: these are all repeats of the same algorithm, with clever U-turns to position a new to be fixed corner to the front/right. When the corner is fixed, go to the next corner, etc...
2 twisted corners next to each other, topcolor facing outward (left and right) holding them to the front/left and front/right (R' D' R D)x2 U' (R' D' R D)x4 U
You can use my BETA solver to get to the first level. But it should be doable. Example method below:
Find the Red White Green corner and manipulate the cube (arrowkeys) to put it to the Left/Front/Down. Red to the Left, White to the Down, Green to the Front.
Find the Orange White Green corner and move it relative to the Right/Front/Up. Use the Right hand algorithm and repeat as many times until the corner is oriented correctly at the Right/Front/Down.
Manipulate the completed corner to the Left/Front/Down (arrow left). Next the Blue White Orange corner, etc...
Right hand algorithm: Left hand: Usage: 1-6 times. To put the twisted corner back to Right/Front/Up (and keep the other 3 bottom corners in place):