For inverse battles, the 4 moves that provide widest coverage are:
Bug - 7 types super effective, 3 not very effective
- Fire, Fighting, Poison, Flying, Ghost, Steel, Fairy x2
- Psychic, Dark, Grass x0.5
Grass - 7 types super effective, 3 not very effective
- Fire, Grass, Poison, Flying, Bug, Dragon, Steel x2
- Water, Ground, Rock x0.5
Fighting - 6 types super effective, 5 not very effective
- Poison, Flying, Psychic, Bug, Ghost, Fairy x2
- Normal, Ice, Rock, Dark, Steel x0.5
Poison - 5 types super effective, 2 not very effective
- Poison, Ground, Rock, Ghost, Steel x2
- Grass, Fairy x0.5
Worked it out with the Serebii inverse type chart.