The Passover lamb (or goat) isn't a sacrifice in the usual sense of that word. Your English translations of Exodus, etc., may talk about "offering the Passover sacrifice", but the Hebrew really just says "kill the Passover". Moreover, no part of the Passover, including the blood, ever touches the altar.

Although the Passover must be killed at Jerusalem, according to Scripture, it doesn't have to be done at the Temple or by a Levitical priest. According to 2 Chronicles 30:17, the pre-exilic, ancient Israelites believed that the Passover could be killed by any ritually clean person, and I don't see anything in Torah to contradict them.