Current Shunt Vs Current Clamp Meter - 15% difference Which one is right?
If you do the math, the Fluke 353 and Ideal 61-746 are within 2.2% error (STD 0.4%).  This is well within the level of accuracy of the machines given (1.5% and 1.7%) with the Ideal always larger than the Fluke.  For me, this correlation means they are the most accurate.

The current shunt is a manganin resistor.  100µΩ, 25W.  Online references state accuracies of ±0.25% (which is down from your 2%).  This should be the most accurate according to workplace lore.

If you look at the errors for both the 500A and 3000A, they start high (10%, 3%, 2.5%) and go low (&lt;0.5%).  This does make sense because it is a resistor and even though temperature coefficient is 0.00001, it will be most accurate at rated values.

Difference between 500A current shunt and Fluke/Ideal ranges from 9-13% lower (3000A from  3-6%).  This tells me there is a systemic error.

The Fluke 45 and Fluke 289 give the same answer.  I'd look at how there are connected to the current shunt.  We are just dealing with 50mV.  Thicker, shorter wires possibly? (Not even sure how it is connected.)

Your problem is you have three answers and you don't know which is correct.  Two agree, but the most accurate (in principle) is always higher.

You need another reference which can be verified some way.  I always try to go back to basics.  I'd go borrow a calorimeter and boil some water.

Edit...

From Calibrating DC Current Shunts: Techniques and Uncertainties


  The five error sources inherent to current shunts are:
  
  1) Connection
  
  2) Temperature
  
  3) Frequency
  
  4) Drift
  
  5) Thermal emf


And...


  Most modern metrology-grade shunt manufacturers are aware of these problems and
  have attempted to design them out of their products.





  Figure 5 shows a metering type shunt highly susceptible to current and potential connection errors


I substitute your image because his was similar.


  The resistance of shunt Manganin rises about 20 ppm (0.002 %) per degree C around
  lab ambient. Applying current causes self-heating, which changes resistance. This change is not linear. Some shunts rise to a maximum resistance at a certain current / temperature level, then fall as temperature continues to rise. 


How fast do you make your measurements?  


  A shunt must stabilize at each temperature / current level. The thermal mass of a
  shunt includes its resistance element, its end blocks, the current cable lugs and connection hardware, and the cables themselves. At higher current levels, a shunt may require more than an hour to reach thermal equilibrium. This is when measurement should begin. 


Not that they say it's not significant for 50/60 Hz.  But you could try your measurements with shielded twisted pair.


  Figure 8 shows ac coupling between current and potential circuits. Coupling can be
  reduced by connecting current leads in line with the shunt and routing potential leads
  together into a shielded, twisted pair extending at right angles from the shunt (green, not red).


Would this account for 9-13% error always in the same direction?  The jury is out, but I'd say yes.  It does give you things you can try.  

I'd pour over that report and you should be able to easily prove that the clamp meters are the most accurate.