that or you can use an app to edit each frame adding a transparrent blurr/smudge effect over the watermark
Sometimes if it was very mild semi transparent channel logo for example this can work reasonably well... but will still be noticable
Sorry I can't remember any specific app name - was a Long time ago I played with something like that.
But am sure many app could do this - kinda like applying your "blurr/smudge" effect as your logo... this is not a image added rather than a photoshop effect applied to each frame of the whole video in the same spot.
Can't be too hard