When you jack up either front corner of the car and listen for any pops or moans.

Tweaking means using a nice size pry bar and leverage against some critical points of the support members to see if they are solid.

Check the tie rods and ball joints along the way and for play in the steering rack ends or its mounting clamps.

The part that worries me is you said your hands get tired. You must be fighting a serious front end or structural issue. The way you wrote it to me it sounds worse than a noise with the brake pedal shaking.

How long have you owned the car and is it a stick shift? I did not catch that part.
The pressure plate assembly has locating pins holes for the pins on the flywheel to fit. It should stay concentric unless someone ignored them. Have you had clutch work done recently?

The clutch cable goes under the brake booster. If, it even can shake the booster, then you would feel that. If the booster vacuum hose became really stiff it could shake the booster too!
I have read where the clutch cable can crack the firewall around where it goes through. Maybe it has spread up and over to the brake booster mounting bolts. See if you can shake the booster housing and the master cylinder unit together!

All of this are wild guesses and your post has intriguing possibilities without more information.
Keep scratching at it and then you will find a place to dig in!

Phil

Phil