Between manual gathering and Crush specialists, keeping White Paint stocked isn't hard. The Crush route is less effort once you've assigned the right Pokemon to the job.

Paint used to color items white

Raw Material
SpecialtyRandom Bonus from Crush Processing
Crush specialty
ShopSmearguru / Palette Town Shop
Bleak Beach (Smearguru hut), Palette Town (Level 5)
ShieldonCrushConkeldurrCrushBastiodonCrushSteelixCrushAmauraCrushAurorusCrushMagmortarCrushRampardosCrush

You can craft Paint Balloon Beige, Paint Balloon Gray and 1 more using White Paint. These 3 recipes are mostly Other items that help fill out your island.

3 recipes
Paint Balloon BeigeOther
White Paintx1Black Paintx1›
Paint Balloon GrayOther
Blue Paintx1›White Paintx1
Paint Balloon WhiteOther
White Paintx2