Below is a list of all the items the Smuggler restocks
He restocks randomly throughout the day,
and it seems to always be 14 minutes past the hour,
The higher the rarity of the item, the less often Smuggler restocks it.
For instance, he restocks Smuggler Plushies and Pearl with a Hats quite often,
but it is very rare to see an Easter Bunny Trading Card or a Glowing Hairy Egg in his shop.
Thanks to Gnomad & Takua95 for the info.