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There is a way to drastically simplify the search for a solution. Use a checkerboard colouring of the triangle grid, as shown in this picture:
The board has an odd number of upright triangles and an even number of inverted ones. The sphinx piece has an even number of each type, so to fill the board the triangle pieece must supply an odd number of inverted triangles. This means that the triangle piece must placed inverted relative to the board. A little trial and error shows that if the triangle is not in the centre, the board cannot be filled, and that with the triangle in the centre only one solution is possible (and its mirror image).
The unique solution:
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