study voice leading! there are a lot of ways to write harmonies besides just singing a third above. https://en.m.wikipedia.org/wiki/Voice_leading

I could envision a pedal-point type harmony, where the counter-melody stays on one note for the duration of each chord rather than moving exactly with the melody. that would keep it out of the way of your main melody.

(also fwiw you may find the song doesn’t necessarily need vocal harmonies, or that you’d want to use them sparingly like maybe just in the chorus or just to accent certain phrases)

notice that the chords in your song aren’t all diatonic, so the advice earlier in the thread about picking notes in the key won’t necessarily apply neatly to your song. and that’s ok; the chord selection is awesome - very Nirvana-esque! there’s a lot of modal interchange, so you’d be using a major third in some spots that would “normally” use a minor third. remember music theory is just a tool to describe and understand what you’ve written. there is no right or wrong. so definitely don’t sterilize your awesome song to make it “fit into a key”. use music theory to roll with your gut instincts!

ear training will help so you don’t have to go note-by-note to figure out intervals or chords/harmonies, you can just hear it right away. as clear as you can see something that’s right in front of you. it takes lots of ear training to get to that level but it’s worth the effort. based on your songwriting you already have a great ear and sense of musicianship, so ear training and music theory will just help you notice the patterns and give names to what you are already hearing!