The white is part of the image so you'll need to remove it. The crop tool is the best option, imo.
I feel that you are exporting a small image in pixels.
That would be the result of anti-aliasing the image.
In the View Menu, Select Pixels as view mode.
Zoom a lot to the problematic edge and you will see this aliasing. Move your object so you align this to this grid.
A quick fix is to export your image at let's say 4 times your desired size. For example, a 300px image can be exported to 1200px and then resampled with Nearest Neighbour to the desired size. This will eliminate this edge completely.
This 4x image can be exported without anti-aliasing.
You can experiment with a 2x or 3x too with the downscaling with anti-aliasing turned on too, this will give you different aliasing levels.