I don’t know the definition of track width, but it sounds like a right triangle problem where the ship length is the hypotenuse and the crab angle is theta.

why not say Cos of crab angle is track width over length?

I don’t think that there is a formula for this unless it is loosely related to the formula for squat.
Which is bollocks.
What are the wind conditions?
What is the swell?
What are the local top and bottom layer currents?
Subsea contours?
What is your hull form?
Trimmed by the head or by the stern?
Where is my gin?
Sorry, last one was just a plea for help:)

While I sort of agree with you, I’m looking for some numbers to make an argument one way or another for certain reasons. Knowing how wide the ship becomes when crabbing is helpful for this.

There is no formula that you can rely on!
It is futile.
If you are part of a management team attempting to lay down fixed rules for Bridge Teams to apply then good luck with that.
If it is a an academic exercise then go out and try it!

For crying out loud, none of those will work in real time conditions, ergo they are useless.
What is the formula for the Master to assess the effect of the wind on the ship whilst coming alongside?

There is no pivot point, just a bit of trig showing the shadow cast by the bow on your wake, so to speak. You can work it out as two triangles with the tips midship, but the answer is exactly the same.