The Draft Orders (Standard) and Accelerated Methods of Custom Pricing do not create variants that could go over your store's variant limit. Visit Switch Discount Methods in Custom Pricing to learn how to switch.
Custom Pricing does not integrate well with other apps that display products outside the store. This is because Liquid code is added to your theme files to hide the customer specific pricing until the customer is logged in.
Generally, when products are displayed outside the store, third-party apps retrieve a range of the lowest and highest priced variants from your store. Some common apps and sales channels that do this are Pinterest, Facebook, and other product feeds.
As these apps do not use the Liquid code installed in your theme, there is no solution to this. Unfortunately, we also do not have control over how these apps pull their data.