You mention that the error occurs when you started to use a package, but you didn't show any package declaration in your code. A first guess would be that the constant is defined in one package, but used in another one, and that you did not import it.

Does anyone have a better suggestion to approaching using constants in this way?

"Better" is often a matter of taste. Here is how I use constants in my current project:

Instead of using use constant, I have installed the Readonly module from CPAN. I declare a constant in one package like this:

package Glogg;
Readonly our $myconst => "goofy";
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and refer to it from another package like this:

package Drull;
print("$Glogg::myconst\n");
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Of course I could also export it if I don't want to refer to it via the package name.