

First floor bathroom fan duct venting options
I’m trying to vent out a ceiling fan through the sidewall. Initially I was going to wall vent in the bathroom but that was before I realized I had lowered ceilings.
The second picture shows 6” clearance. The stud is indicated in red. The second floor joists are above the old ceiling (I checked with a bore scope).
The ceiling frame is a 2x4.
My question is can I make a hole for a 4” duct to vent out in this space? Do I need blocking to fortify the ceiling frame?
Also I might have to take a piece of the wall frame too since 2x4 only gives 3.5”clearance. I’ll need another 1/2”
Or do i just do a through the wall vent?
Any suggestions or advice appreciated