How can I apply discounts (via third-party apps) to customized products in Shopify?

You don’t need to use any 3rd party apps to apply discounts, as Shopify doesn’t allow dynamic price modification in the cart. So, the best solution is to use InkyBay’s built-in Quantity Discount feature.

InkyBay creates a new product with the final price and image when customers place orders after customization. These new custom products created by InkyBay don’t have the same name and tag anymore. That is why the discount codes don’t work when you apply the discount to specific products or product tags. 

InkyBay's Quantity Discount feature allows percentage or fixed-amount discounts based on product quantity and applies to the base price, design costs, or option prices. Check out the video tutorial on setting up quantity discounts.