Affiliate Direct Links are designed as an instant-checkout Buy Now button, so it's not possible to combine those with regular cart buttons for buyers to build a multi-item cart order, as you have seen. You would want to use Direct Links for revenue-shared products together with Buy Now buttons for non-shared products on your site, so buyers would have to make a-la-carte purchases for one product at a time.
Really, using our affiliate features to calculate revenue splits with suppliers is sort of a hacky kludge approach, using a tool for something it was not designed to do although it can kinda be made to work that way clumsily, like using a wrench as a hammer.
Also bear in mind that it does not actually split payment to pay the supplier-affiliate their cut on a per-transaction basis. You would receive every payment in full, and the affiliate's share amount gets added to their monthly lump-sum commission total, so after the end of each month, you would need to use the Seller Admin > Pay Your Affiliates function to obtain the masspay.txt file we provide, which you could submit to PayPal to pay out commission shares to all your affiliates for the completed month.