Here are some thoughts about the name:
* "Two Be Bar" might refer to a ranch or breeding program. If you're naming the horse for a specific place, this works well.
* "Baker" could be a reference to the horse's color (bay) or a more personal meaning.
Overall, it's a fun and creative name for a Quarter Horse. I think it's a good choice!
Do you have any other details about the horse that might influence the name?