Hey there @korinathekoala -- for Jade Beads I wouldn't say there's a sure way to get them every hoard, but one way that you can raise your chances of getting them is by using Special Ingredient Potions on your pet's hoard. These will cause your pet to find at least one special ingredient, but not always a Jade Bead. Potions can be made in the Potion Brewing Station, and the recipes can be viewed by expanding the "Show All Recipes" menu at the bottom of the page.

Edit: I just remembered that Jade Beads have a 30% rarity, which means you can further boost your chances of finding them by using Item Rarity Excluders (another type of potion) alongside the Special Ingredient Potions. If you do end up making these, you may want to plant a bunch of Voxel Seeds in your house, since the recipe requires 8 Voxel Fruits.

Shiroking is one of those pets that takes some patience to get, so I wish you luck!

Please let me know if anything I said was confusing or if you have further questions!