Is there a Scripture passage that says that the Priest had to be cleansed from sin before he could approach the mercy seat of the Ark?
Yes, Lev 16:.2-4 “Tell your brother Aaron that he is not come whenever he pleases into the sanctuary, inside the veil, in front of the propitiatory (mercy-seat) on the ark; otherwise, when I reveal myself in a cloud above the propitiatory, he will die. Only in this way may Aaron enter the sanctuary. He shall bring a young bullock for a sin offering and a ram for a holocaust. He shall wear the sacred line tunic, with the linen drawers next his flesh, gird himself with linen sash and put on the linen miter. But since these vestments are sacred, he shall not put them on until he has first bathed his body in water.
v.6” Aaron shall bring in the bullock, his sin offering to atone for himself and for his household. vs. 11-14 “Thus shall Aaron offer up the bullock, his sin offfering, to atone for himself and for his family. When he has slaughtered it, he shall take a censer full of glowing embers from the altar before the LORD, as well as a double handful of finely ground fragrant incense, and bringing them insided the veil, there before the LORD he shall put incense on the fire, so that a cloud of incense may cover the propitiatory over the commandments; else he will die. Taking some of the bullock’s blood, he shall sprinkle it with his finger on the fore part of the propitiatory and likewise sprinkle some of the blood with his finger seven times in front of the propitiatory.”
This is how Aaron would atone for his own sins and that of his family once a year on the Day of Atonement, otherwise known as Yom Kippur. The other sections of this chapter speak of the two goats, one which was for the sins people of Israel and also for the cleansing of the tabernacle. The other goat is known as the scapegoat. Aaron would lay both his hands on its head and confess the sins of the people of Israel. It then would be let loose into the desert.
Hebrews 9:24 “For Christ did not enter into the sanctuary made by hands, a copy of the true one, but heaven itself, that he might now appear before God on our behalf. Not that he might offer himself repeatedly, as the high priest enters each year into the sancturary with blood that is not his own, if that were so, he (Christ) would have had to suffer repeatedly from the foundation of the world. But now once for all he has appeared at the end of the ages to take away sin by his sacrifice.”
So, eventhough Yom Kippur is six months after Passover in the Jewish liturgical year, Jesus Christ is both the Passover and the Yom Kippur sacrifice. When the veil was rent (as depicted in the gospels), we are informed that Jesus Christ entered the holy of holies in the heavenly temple once and forever with his own blood as “the propitiation for our sins, and not only for our sins, but for the sins of the whole world.”