Sweetie, 3-1/2 months is very little new growth ( about 1-1/2 inches). Your mom may not go for that. You may get more support in transitioning. Have you researched that option? If you can prove that you can take care of your hair while doing that, your mom may let you cut it after you gain some length.
Good luck and congrats on your decision to embrace your natural hair. Natural hair is soooooo versatile. Maybe If you can show your mom what some of the ladies on YouTube who have similar hair to yours are doing with their hair, it will help some.