A bitless bridle does not equate in any way to a halter with reins attached. It uses pressure points -- if you want to try it, you need to actually have a bitless bridle.
That said... maybe there is another strong rider on a calm horse that would be willing to ride with you? How can such a large show horse barn not have an arena or round pen? Are you comfortable riding anywhere with this horse? On any horse? Maybe you need some time on a steady, calm mount to get your own confidence up. If you can ride another horse for a while, I would just continue lots of ground work with your 4 yr. Old and improve your own skills for a while.