Round bowls are tricky when you need to do certain things to them at times.
A reflection usually happens on glass tanks, and when the aquarium light is on and the room light is off.
Does he circle the tank? Or is he flaring/attacking the side?
I have a couple round tanks for my two baby bettas, and they will swim continuously around it, and sometimes will fidget. They are plastic, so they don't really show a reflection when the room light is dim, or if at all since uncertain if the plastic tanks can actually reflect..
If he's attacking the side of the tank, eventually he will calm down, if he is just circling the tank then that just may be him liking how the water feels on him when swimming quickly next to it. A lot of times you will see a betta pace back and forth when in a rectangular tank, but in a round one, there are no sides to break up the pacing so they just continue to swim. It's not harming him, and he will stop when he tires.
As for the not eating, how long have you had him? Not uncommon for newly adopted bettas to take a week or more before they eat healthy and hearty. As long as he is eating then he is getting nutrition. He won't let himself starve to death.
What are you feeding and how much and how often?
But honestly, he isn't chasing his reflection so much as he is exercising and building up his muscles from being in a breeder's jar/cup for so long. A 10 gallon would be great for him, or divided for a few of them :)