The Filter Alpha is very much like doing surgery with a chainsaw. It checks to see if the requisite number of pixels are set and uses that to determine whether or not to cut off the block.
The black background image is an unfortunate reality but there's some things you can do to cleanly remove it but it gets more complicated.
What I would recommend doing is finding some imagery to use a background map. You can use OSM Raster Tiles, for instance. The source data can be anything as it's just being used to replace the black and give you a point of reference. You want to make sure that they cover your region up to the extents of the data you care about.
If you get a suitable boundary around the data, you can use the Filter Alpha option, the extents options, or the Filter Shapefile option.
Extents allows you to specify the outer extents of where you want the raster data to appear on the card. This would be in the same projection as your source.
Filter Shapefile allows you to provide the exact location of your outputted raster files as polygon vector data in a shapefile.