I think that a new hole through the floor or bottom plate of the wall is the solution. A 4" hole in a 2x6 bottom plate is no biggie, just needs nail plates. A 4" hole in a 2x6 stud ranges from sort-of-harmless to an imminent structural danger depending on why that wall uses 2x6 studs instead of 2x4. If they're thick for load bearing, you'd risk a collapse of some sort. If they're thick to contain plumbing and pipes, it's sketchy and inadvisable but viable in a pinch. Be very cautious with your assumptions here.

Ideally, your large pipes go straight up or down within walls. And in floors or ceilings, they should go between joists unless they can cross over in the attic or under in the basement.