Is it permissible for a person afflicted with a disease that causes pus and blood to constantly exit from the anus, thereby contaminating their undergarments, to pray, given their struggle and exhaustion from changing clothes for each prayer?

The fundamental principle is the obligation to purify clothes from impurities that affect them, and these are not excused except out of necessity. If it becomes excessively difficult due to the continuous discharge of blood and pus, there is no harm in adopting the view of the Malikis who excuse such impurity due to hardship, because it then becomes like an enduring ritual impurity (hadath mustankah) that affects the body or clothes. The more cautious approach is to wash the impurity even with incontinence.