Why exactly does peritoneal dialysis need a greater amount of protein?

Protein loss. Due to its anatomy the natural peritoneal membrane used in this kind of dialysis is a leaky one that allows passage of proteins especially Albumin to the fluid that will be discarded in the bag after dialysis so a good diet with proteins is a must.
PD. Dialysis via peritoneal dialysis promotes loss of amino acids and loss of large protein molecules. In general a 24 hour session of PD will cause of the loss of approximately 8-15 grams of protein and amino acids. If a person acquires peritonitis, then more protein will be lost. Hence extra protein intake is essential.
See below. There is some protein loss across from the blood into the peritoneal fluid used to eliminate waste.