About a year and half ago I bought a Soma Rush thinking it would be an upgrade from my Mark V. Immediately I noticed how harsh it was. At first I thought it was the bars, Deda Pista. I know they're super stiff so I figured that must be it. I swapped them with the bullhorns I had on my Mark V. but didn't notice much of a difference. My next guess is the fork. I know my Mark V. was WAY smoother when I swapped out the stock steel fork for a carbon fork. Has anyone done this on a Rush, and if so what are the results?

Do you have the Soma straight blade or the curved forks? The curved forks on my Rush give a pretty smooth ride, especially for a semi-track bike. With a 1" headtube and a 38mm rake there's not a whole lot of options for carbon forks. Wound up makes 1" forks in 35mm and 40mm, they are pricey and known for being stiff though.

I have the curved blade fork.

IMO a rush is an upgrade from a mark v but My opinion means nothing. For most people who want track bikes stiffer is better. Maybe you could have gone with a road bike?

if you just want it to ride nicer on the street get a carbon fork like the forte precision or whatever.

if you want a comfortable bike, get a new bike. the rush is a great bike but it's going to be stiff no matter what, you can only dampen that stiffness with bigger tires and cushier forks.

I realize it's going to be stiff. I have many bikes and the Soma is not the stiffest. It is, however, the harshest. That's what I want to change.

It could be a lot of factors, or a combination of them to be honest.
It could be your wheels are stiffer than the wheels on your MKV. Maybe swap those over and see.
Your hbar/stem could be stiffer. (edit: Nevermind, just saw that you changed handlebars already!)
Seems trivial but tire pressure could be higher on your Rush, or running smaller tires at the same pressure.

Or the frame is just stiffer! Ha ha. My Rush is soooo much stiffer than my classic Benotto road bike.

You could try a carbon fork/seatpost if you could get your hands on one for a day or two. Before you make a move on a new fork I would try checking the other factors, could work!