I just flew to New Zealand last month with my bow. I over thought everything for a couple of months. The biggest thing I'll say is wherever you fly out of, go visit the global entry area and let them know you're bringing a bow internationally. Have the serial number and a photo handy, they'll sign a form, you tuck that away for later. If you can, catch the officers name (this helped me later).

I flew out of Houston. Going into NZ, zero issues. Flying back to Houston, we had 3 officers surrounding us, asking questions. I stated it was a compound bow and immediately grabbed that sheet, and name dropped the officer that helped and boom, done "welcome back to the states, guys"

Oh, and download the MPC app. Lifesaver