The problem is that you have made the gradient blue-transparent rather than blue-white. The white isn't actually white, but transparent against a white background (the background of the post on the forum). Since gifs only have on-off with regard to transparency, it turns all mostly-transparent pixels completely off when trying to convert it to a gif.
If you make sure it's actually white in Inkscape it should work right.
GIMP should work fine after that. But if you are comfortable using a terminal then ImageMagick would probably give you a much faster workflow.
This example turns the pngs into gifs while replacing transparency with white (flatten). Then it animates them in the last row.
$ convert -flatten ghost1.png ghost1.gif
$ convert -flatten ghost2.png ghost2.gif
$ convert -delay 30 ghost*.gif animated-ghost.gif
But you're better off doing the transparency->white conversion in Inkscape, since this turns the entire square white and not just the gradient.