It's not a sport cap, but have you tried a splash guard? https://www.amazon.com/BottlePro-SplashPro-Splash-Guard-Flasks/dp/B087H5QVCC

Not sure how well that'd work for you, but at the very least it would limit spills. I use mine all the time while driving and have no issues with spills.

(full disclosure, I'm the designer and seller of these splash guards...sorry for the shameless pitch, but hopefully this helps)