Regumate is what you give a mare when she needs Progesterone, a hormone that maintains pregnancy or a CL. We gave it to mares to help them keep an early pregnancy (until the fetus made its own progesterone) if they've had issues in the past or there's a risk of abortion. I think it's also given to mares to keep them from cycling during the show season, but don't quote me on that.
AI is considered far less dangerous for the stallion, mare, and handlers. With AI it's very simple, the semen is drawn up in a syringe, the tip of the syringe goes into her cervix, and then the semen is injected in. Live cover means contact between the mare and stallion, which can sometimes get a little iffy because they can both become unpredictable and lash out. It's true that rectal exams can hurt the mare when done improperly, which is why vets are only allowed to do it. (that's how it was at my internship anyway, no techs or interns were allowed EVER on client mares) I don't know what the price difference would be between AI and live cover, you'd probably have to call whoever will be doing your breeding to check prices.