Aseprite sets the first colour in an Indexed document’s palette as transparent on layers (but not on the Background layer, I believe). Solutions:
A. Use RGB mode, and your palette will act more like a set of swatches. Indexed Mode in Aseprite in general seems a little funky, and you can still easily keep your image to only the colours from your palette even in RGB mode, so there aren’t a lot of benefits to using Indexed Mode.
B. Add a transparent colour to your palette and move it to the first slot of the palette. This is required because each layer is Indexed too, but layers require a transparent colour to be useful, so Aseprite just picks the first colour to act as the transparency. By giving Aseprite that extra colour to use as the transparency, your other colours will be freed up from acting as the transparent colour.
Edit: Reading your post more closely… it’s also possible that Pivot Animator might be loading the sprites weird? If the above doesn’t help, double-check your export options or save as (RGB) PNG.