Did a scope expansion for DC energy measure. So not only dealing with just DCI measure, but DCV, time, simultaneity of measure, and more. You are going to have so much fun.
Get a danisense DCCT, can't beat it. Like 10 ppm, in series with your shunt, and compare. The danisense guys, were some of the best from LEM. I've tested both the ultrastabs and the DS series, both are incredibly good, the danisense edges out the LEM IMO. But likely, given the level of your equipment, it wont matter, so get what ever is cheaper. They also have great acd/dc properties, so could open up your lab to ac measure later as well.
As for the thermals of the shunt, you either go one of two ways:
You energize to current level, hold, and stabilize temp. What I have found with this is that, shunts can take a significant amount of time to stabilize to a level of very low variability. Like over 30 minutes. Then doing that for a number of test points, make this method incredibly time consuming.
Or either swap connections, or shut on off, and negate thermal EMFs. I do something like 5 second measure off, 5 second measure on. 5 second measure off. Average before and after and subtract from measure. 5 repeats. Point being quick measures to mitigate self-heating. high dTemp/dTime kills ability to temp correct, and will throw erroneous results.
Also looking at your pictures, need better metal matching. The owner of Ohm-labs, Jay has great material, and is a good source of knowledge when it comes to all things resistance and testing. Sadly he is mostly retired, so he may not be as available to chat as he used to.
For the thermal characterization, you need a more homogenized environment, like an air bath. But you can go more junkyard approach, with a box and a fan and temp sensor. If you are curious to see what type of equipment it takes to do this work, what I typically do is research who the players are in the test equipment field, then I look for the ISO17025 scope, and then find the measure, and measure range to find the calibrating equipment.
There is so much more, but my brain's quite scrambly from not sleeping. Relatively new father (1y.o.), much harder than uncertainty budgets.