How would I add a Constant to another class?
The short answer: you cannot add a constant to another class.
However, the constants, as their name says, are just names for values that do not change during the execution of the program. This is probably the reason they are all public and their visibility cannot be restricted.
All of them being public, the only difference between the global constants created using define() and class constants is the syntax on creation and access.
The complete answer to your question is: you cannot add a constant to a class during runtime (from the code of another class or even from the code of the class itself) but you also don't need it. It's enough to define() a constant or declare it as const in another class.
The only way to add a constant to a class is to modify the definition of the class (using an editor, of course).
MyClass1::CONSTANT_FOO.... and note that you should be setting a value in your constant definition