You could try Mare Magic or another herbal remedy to try to balance the mare's hormones. I believe some horses have pain with cycling just as humans do, and some horses deal with pain poorly by taking out their angst on whoever or whatever comes near.
Sometimes horses have polycystic ovaries like humans, which causes their cycling to become irregular. I've known a couple of people who have spayed their mares, but in order for it to be effective you need to do a full hysterectomy. That is very expensive, somewhere in the range of $2,000.
Putting the mare down seems extreme for something that is almost certainly related to a physical issue or else related to her young age and lack of experience. Many people, including myself are more than willing to deal with mares that become temperamental when they are in heat, and some mares get better as they age. One of my mares had serious issues when she was in heat until about age 16, then it began improving. The other one is just temperamental all the time, LOL.
If your mare is registered and has good bloodlines, consider selling her on the Arabian Horse Association website classifieds or Dreamhorse. She may end up doing better as a broodmare, or after having a foal, or just with an experienced mare and Arabian owner. Quite possibly she would make a good endurance horse, and lots of exercise can help with hormones too.