If you *think*
that your horse respects you on the ground, but yet he refuses to do what you ask while you are riding, then you actually have NOT taught him who is boss.
Whenever you are around your horse (not just "round penning" him) you are showing him how he can handle you. Does he get into your personal space, uninvited? Does he lead you, versus you lead him? Can you lead or send him anywhere, or does he go only where he wants to? Does he willingly pick up all 4 feet? Does he drop his head for you when you ask? Can you move any part of his body from the ground? If you have a problem with ANY of these things, then your horse is the boss -- not you.
Also, it makes a difference what you do when you "round pen" him. Do you just make him go in circles, at the speed he chooses? Or do you fill it with numerous direction changes, stopping, backing, speed changes, etc and making him listen to your body language?
It just doesn't make sense that he would be perfect on the ground, and not perfect when riding, so there's got to be some holes in his ground training and respect training somewhere.
Also, you did not answer: Have his teeth been examined by a trained equine dentist?
If not, he may have a mouth issue that causes him pain with the bit.
Even if you can't afford to all-out send him to a trainer, how about taking a couple of lessons on him? It certainly never hurts to have a trained professional watch how you ride. Everyone can always improve on something.
How about posting a video for us? It's hard to "guess" what might be wrong just through words, since timing
is so very critical when getting a horse soft in the mouth.
As far as barrels go, which you did say already, but you need to have this horse's body 100% broke before introducing barrels, and be able to control every piece of his body at every speed. Check out the barrel sticky for more info. Barrel Racing Exercises and Drills.