Eureka! After beating Google up for days now on this I found a very simple solution that seems like it would have so many other applications I am surprised I have not heard of it before. Setting an Environment Variable allows you to pass values from within a user, process, or machine context in an extremely simple way. This seems to me to be so much simpler than using and maintaining nondescript callcontext fields. Big thanks to Adam Ellis for posting this at Ginger Help. Final Code snippets below:
In the launching dashboard customization:
string qtyPassed = edvPutAwayView.dataView[args.Row]["UD105_QtyPassed_c"].ToString();
Environment.SetEnvironmentVariable("QtyPassed", qtyPassed, EnvironmentVariableTarget.Process);
…and in the receiving (Inventory Transfer) customization:
string qtyPassed = Environment.GetEnvironmentVariable("QtyPassed", EnvironmentVariableTarget.Process);
Just repeat for as many variables as you need.