Most importantly, avoid accented dissonance.
Next, I think harmony is going to be your friend here. The backing track's harmony is rather sparse and therefore somewhat ambiguous, so there's plenty of room for you to steer the mood with how you choose to harmonize the melodies.
I know that the relative major is Bb Major (harmonic), but that
doesn't really do any good because it's just the same notes in a
different order. Is it even possible to stack something on top of a
minor scale/key that will help it sound "happy"?
Keys are more than just collections of notes. Even though they use the same notes, G Minor and Bb Major sound different. Yes, you can play Bb Major over G Minor, and yes, this will make it sound less dark.
Here are some harmonic ideas which might help. These work because they are diatonic, ie native to G Minor, so they won't sound too exotic or dissonant.
- Use Bbmaj as your tonic chord instead of Gmin. If you're harmonizing a G or Bb, use Bbmaj. "But Bbmaj doesn't contain G" you say. True, so it's really more of a Bbmaj6. You can assert this by keeping Bb in the bass.
- Use D7 as your dominant chord. D is the dominant of G Minor anyway, so nothing strange here. But I specifically recommend making it a dominant 7th chord, which to my ears at least make it sound brighter. If you're harmonizing a D, F#, A, or C, use D7.
- When neither Bbmaj nor D7 works, try Ebmaj. This will harmonize the Eb which is featured in the backing track's melody. It will also harmonize G and Bb, and can be used instead of Bbmaj if that sounds better to you.
The theoretical basis for this is chord function and chord substitution. Basically, there are 3 functions: tonic, subdominant, and dominant. In any given key, each function will be strongly associated with a particular chord. In G Minor, these will be Gmin, Cmin, and Dmaj, respectively. This is a basic i iv V progression, and lots of music uses it. But it's kind of vanilla. Other chords can substitute for these chords, serving the same function while providing a different flavour. The simplest chord substitution is to use the diatonic chord found a 3rd away, either up or down. This is called a mediant relationship. My suggestions use III in the tonic function (instead of i), and VI in the subdominant function.
- tonic: I (vi, iii)
- subdominant: IV (ii, vi)
- dominant: V (vii*, iii)
- tonic: i (VI, III)
- subdominant: iv (ii*, VI)
- dominant: V (vii*)
(Note there is some overlap between the categories, so context is important.)
You could go for more exotic sounds by borrowing notes and chords from other keys. This is called chromaticism. Technically, in G Minor, D7 is borrowed from G Major, but it's so common that we don't hear it that way anymore. But make note of how taking the natural minor scale and raising the 7th scale degree has made it sound less dark. And if you raise the 6th scale degree too -- which makes the melodic minor scale -- it gets brighter still. You can carry this idea further. By making flattened notes natural, or natural notes sharp, you're brightening the sound. There are 2 more or less equivalent ways to think about this. You can either think modally, recognizing that some modes are darker than others, or you can think in terms of key relationships and the circle of 5ths. If you're more comfortable with the circle of 5ths, you want to borrow notes/chords from sharper keys, ie those that have more sharps (or fewer flats). If you're more comfortable with modes, borrow notes/chords from the less flat/more sharp modes.
the modes from dark (most flat) to bright (most sharp)
- aeolian (aka natural minor scale)
- ionian (aka major scale)
This idea is fraught with peril though, because as you add chromatic notes (ie notes foreign to the key) you are more likely to run into dissonance, so be careful. Also keep in mind that notes/chords borrowed from distantly related keys will sound more foreign than those borrowed from closely related keys.