Sending a big band score to someone who needs a score for his band?

First, I AM NOT A LAWYER!

Nonetheless, your work is covered by copyright as soon as you have written it down. No matter whether you write to the Library of Congress or send that unopened, letter by certified mail.

This, however, does not protect that copyright. If someone wants to steal your work, then you have to pay a lawyer to sue them. That is the protection. Having the score registered with the Library of Congress or having that certified letter, simply makes it easier to win that court case.

As I said above, I'm not a lawyer. If you are concerned (and I would be), you would be wise to consult a lawyer to get definitive information and advice.

As to sending it to your big band leader (assuming you decide to!), export as .PDF and email them the .PDF file. Or export as .MP3 and email that. (With a .MP3 they probably can't get your actual score pulled out.)

Good luck!!!

[A few moments later] And the description above consists of my understanding of US law. If you're anywhere else, your local copyright laws may be subtly or wildly different.

You wrote:
Is there a way to send a big band score to the leader of a big band that wants to hear my score...?

You have options as to how deeply you wish to dive to produce results that are satisfactory to you.
MuseScore can export your score as audio - for example, .MP3 is already mentioned.
I would add that for audio export you should pay attention to score elements such as dynamics, along with dynamics changes like hairpins (cresc., dim.); and tempo with changes (rit., accel.).
Also, consider sounfonts:
(https://musescore.org/en/handbook/4/soundfonts#Using_sounds_from_a_Soun…)
...and Mixer settings like panning (listen with headphones) to get an audio file that "plays" your original score to your liking.

Finally, there is the issue of (lossless) audio quality,
See:
https://musescore.org/en/handbook/4/file-export#Audio_formats
and for creating "recording studio" audio:
https://musescore.org/en/handbook/4/working-vst-and-vsti#Introduction_t…

Good luck with your score!
P.S.: While not a "big band" score, here's audio which a MuseScorer produced years ago - for his composition dedicated to the MuseScore dev team:
https://www.youtube.com/watch?v=Ojqk3EhlWv4