...
One circular eccentric disk, mounted on an offset pivot pin so it can
be centered or swung out as needed. The clamp for the swinging side
might have to include a custom stepped bushing to withstand the
torque, rather than just a bolt that holds by friction, but its
surface finishes aren't critical like the eccentric's. ...
...

I haven't got this for certain.
Not grasped the idea for sure, yet.

Forces would be in Tonnes to tens of Tonnes.
The size of the eccentric - both what it will withstand as a bend or
shear, and preesenting enough bearing surface for the bearing to take
the load - can make these things quite big. Maybe 200mm diameter with
190mm shaft for 10mm "throw", for some enormous tests in high yield
steels.
As I calculate / estimate as best I can.

If it came to making one of these things, there would have been a lot
of proof-of-principle with "testing lab. scale" samples, and the
stakes would be quite remarkable by that stage, if we got there.

The main point is to have plans in place.
In a political world, you have to have everything covered, so every
interjection, objection, etc. is smoothly put in its place.
As I've experienced.
I've certainly had the skill of predicting what the ploys might be
tested. Having big efforts to derail the plan slapped down in
seconds.

So it's about being able to see a way ahead, far along a perceived
path. What would actually be going on, where you would actually be by
then, what method you would use given experience had by then but ahead
of you now - that might be a different story.
But for now - you are "covered"...

With the beam tests, you can tune the testing force by moving the beam
end supports in and out - present different spans. So one fixed throw
/ movement drive would cover all purposes.

For the "hydraulic inner fatigue test", I think different diameter
"spuds" sliding in and out of the fluid volume, on a fixed eccentric
drive, might be easiest?