I've been trying this with my bass this morning, and I find that the breathed note gets drowned out by my bass, making it hard to compare the note I'm thinking/singing, with the one I'm playing.
Is that okay?
Also, I will say that my signing and vocal pitch reproduction is bad (I'm working on it), but because I can't really hear this breathed note, it's hard to tell if I'm even doing it in tune.
If you are practicing alone, just sing out loud. The pseudo-whistling thing was for privacy in the classroom. In other words the professor wanted us to be quiet so we didn't bother each other.
Ideally you should eventually learn to hear pitches and chords in your head. The pseudo-whistling can be a step in that direction. Practicing Solfege and other voice/ear training exercises with a piano is probably the best thing you can do...but not everyone has interest in or access to a piano.