None of the current interpolation tools do "round the world" interpolation. I.e. the search neighborhood ends at 0 and at 360. Think of it as a rectangle 0 to 360 and -90 to +90. The points on the far left, circa 0, are not considered neighbors of the points on the far right, circa 360. Also, the Arctic Circle is a rectangle! Also, a cell size of 0.04 at the equator and a the poles, what does it represent?
If you make the cell size too small then you might encounter the SplineWithBarriers size limit (16500; 16500 columns and rows). However, a cellsize of 0.04 degrees only yields a raster with 9028 columns and 4358 rows.
To do this type of interpolation you'll need to interpolate in blocks and Mosaic the output. Also best to work in projected coordinates and not in Geographic coordinates.
-Steve