I have two lowered Mustangs and don't use a bump steer kit on either.
Don't feel the need. Both cars are very stable.
I've never driven a car so equipped so have no feedback to offer on bump steer kits.
What I do know, from experience, is that all Mustangs, except for those special editions where attention was paid specifically to suspension such as Shelby's, Boss's or Performance Pack editions, have just average handling from the factory.
They have body roll and vague on center feel.
They nose dive or squat under braking or acceleration.
They have a floaty front end and/or too much rebound.
Like riding a boat on swells, or and old Lincoln with under inflated tires.
With the age of your car dampers could be getting tired, leading to momentary loss of tire contact with the road on the rough roads you describe.
Each time there is such contact patch loss the car's tracking of you travel direction is lost or disturbed.
This gives an erratic drive.
This could be the feeling you are describing.
All of the above is why I tell enthusiasts that good suspension work on a Mustang yields as much Wow factor as adding a bunch of power.
Many or most don't believe me because they simply "don't know what they don't know".
I can use my low mileage 2015 as a recent example.
It had a flaoty front end and bouncy rear end from too much shock rebound.
It was not a fun drive.
I cured that with a conservative drop using minimum drop BMR springs and Ford Performance track dampers.
It's a whole new car. Fun to drive. Totally confident handling. Amazing difference.
To cure any residual body roll, I also did Eibach sway bars.
I would suggest that your car could use a similar transformation.
A set of conservative drop springs from Eibach, Steeda, Ford Performance or BMR.
You can't go wrong with any of these. You pick.
And a set of dampers. You can't go wrong with Bilstein.
Ford Performance, Eibach or Steeda would also be good options.
Don't rush for the sway bars. Those can be done separately.
Wait to see what results you get from the springs and dampers.
You won't believe the transformation.
It will make you "Love" your Mustang all over again.
That feeling you had when you first got it. Remember?
Your internal reaction will be: "Wow, I should have done this sooner!"
I had the same issues with my 2005 and 2006 solid rear axle cars previously.
Same cure, springs and dampers.
The first time I did suspension work, I got that feeling of "Wow, I should have done this sooner!"
That's how I know.
The Mustang is a relatively inexpensive car for its sporting nature, especially in GT form.
Similar power to Europeans costing two, three or four times the cost, or Corvettes and other similar performance cars.
In order to keep the cost low for that much power, something has to be left on the table.
On the Mustang, the suspension is one area where a significant amount is left on the table.
Respectfully submitted and based on first hand, personal experience.
Regards, and good luck with your search for a solution