How to track purchases with Facebook Ads + CJ Affiliate?

Background: I run a business that makes money off affiliate marketing. I use CJ Affiliate to connect to the affiliate business in question. What's the best way to track whether or not a Facebook ad caused a user to click on an affiliate link and buy the product?

The problem is that I lose track of the funnel at "add to cart" which I really just have mapped to the "Buy on SomeAffiliatePartner" button. Since I don't have access to the SomeAffiliatePartner website, I have no way of knowing if a purchase was actually made for a specific link click besides looking at a dashboard manually on my CJ Affiliate admin console to see how many purchases happened that day. This makes it so I can't optimize my FB ads on purchases, rather just on "add to cart" which isn't ideal.

Has anyone ran into this issue before and how did you solve it?

Cheers!