How is $dt unity at the end of the subroutine and a well-formed string in main?

Since we haven't an explicit return statement, the sub returns the return value of say.
Either remove the say, or add return $dt;

How do I fix my duration calculation to reflect the difference in two dates?

After having jd2dt returning a DateTime :-) your $dur is a DateTime::Duration object.
This doesn't have a stringification method, but you can have it back in various units (with limitations, see its ->in_units method).

P.S.

I have to wonder out loud how you were able to cobble it together from scratch so quickly.
… Julian dates aren't the stuff that most people can wrap their heads around.

But, but, but, Julian Days have been invented precisely to make date calculations as easy as working with decimal money!

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