Yeah the extra transparent space in the image can cause a big headache.
Ideally if you can in your desktop app:
1. save the image at its largest sitze
2. crop the image down to as close to the pixel space eg. see green border here:

3. Then save the Image at the desired size you want it displayed eg. 300px wide
As a general note, when saving new logos, each new logo you upload should have a different file name.
As WordPress caches the logo, and if you upload a new logo with the same filename as the old version, then WP may just use the old one.