Codecs changing file demensions after encoding

Quick rundown of the situation;

A client of mine is a minor league hockey team. They recently got a bunch of LED screens installed around the arena that they use to run sponsor ads throughout the game. One of these LED screens is a half ribbon running around half of the arena underneath the suites and has a resolution of 12288PX by 64PX. The client is requesting h.264 exports as everything else produces file sizes too large that cause their systems to lag. However, whenever I attempt an h.264 export, it compresses the file in a way that causes the file to be shrunk/ and adds black bars to the top and bottom of the video.

Anyone have any solutions that will allow me to get smaller file sizes without running into this problem where these black bars are sandwiching the actual content?

Thanks in advance!