Vent Question

Hi guys long time reader first post.
I am 4 weeks into flower and having some smell issues.
I ordered a 4" carbon filter with a 4" fan
My box is 2.5' x 2.5' x 4.5'

I have been having great success as far as the grow is going however I have a set up that will be difficult to get the carbon filter into the box. I have attached a drawing (sorry i am not a paint pro) of what my room looks like.

The red is what I can do with what I currently have. The blue is a plan that would require 2 more holes cut into my cab (its an old dresser of some sort) come out and back in; in order to fit the hose for the filter. Then run it through my light and out the old vent hole.
So my question is will the red drawing be sufficient to scrub my air by pushing into the filter. The light is literally as tight as it looks in the drawing. Or would I have to make holes in my box and pull air from inside out?

Any help is appreciated.
Cheers!

Here are a few pictures of the stinky ladies
They are fem purple kush from marijuana-seeds.nl

Thanks for the info and Enjoy!

Pull air from inside out, use a passive intake. I like to keep the filter separate from the cool tube setup.
You can even keep the fan and filter outside of the room just make sure your connections are 100% sealed.

Any one have some advice? Stuff arrives tomorrow. I will figure out something I was just seeing if anyone has had this problem.

Listen to J420....pull the air outside, scrub the air outside the box, and use passive intake.

So I would be doing my original plan? Or are you saying leave the 400 watt cooled the way it is and do a separate outtake for scrubbing?

Trying to incorporate the cool tube will be the hardest part - try to keep the filter external