There is a pill your doctor can prescribe that will kickstart your period. It is not birth control, but an ovulation-stimulator.
I used to have problems ovulating so I didn't have as many periods as I should, and I used this drug once, but I forget what it's called. I have to take birth control now to have periods, but if you don't want to do that, you can also just take that same drug four times a year. You'd only have four periods per year, but if you don't want to be stuffed with hormones, it's an alternative to bc.
I have never heard of using birth control to force a girl to begin having her cycle.
And don't worry about it! Sometimes girls don't start their period until age 15 or later. Your doctor will let you know when to be concerned, but I would recommend going through with the blood test since your doctor suggested it.
It is very important not to take hormones for very long. You will need them later.
I know you want to move on and catch up to your peers, but I really think it is likely because you have low body fat. To add body fat, I suggest just adding two pieces of pie for dessert until you have put on just a tad of fat. Pie worked for me when nothing else put weight on me.