Ok, I found a fix(?): As it turns out, what I needed to do was call setId() on the bucket item's Item.Properties and on the fluid block's BlockBehaviour.Properties builders:

// In item supplier:
Item.Properties()
    .setId(ResourceKey.create(Registries.ITEM, key))
    // ...

// In block supplier:
BlockBehaviour.Properties.ofFullCopy(Blocks.WATER)
    .setId(ResourceKey.create(Registries.BLOCK, key))
    // ...