When I played LL, we used both MLB and non-MLB names. For example, we had the Astros and the Cubs and the Reds, but also had the Rebels and Colts and Fireballs and Bucks. In my LL career, I played for the Mets, Rebels, Astros, Tigers, Tigers, Tigers, Mets and Mets. Of course we all had sponsors too. One of my favorite years was playing for the Astros. That was the first year we wore real uniforms. We had old flannel hand-me-down suits from the late 60's, but we wore real stirrups. That's when I fell in love with strirrups. It's not a real baseball uniform without them.
What I don't like is this trend to call your team some non-specific singular noun...like "Heat" or "Blaze" or "Terror". It just ain't right.