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.
My hair: thick, medium/coarse, high porosity in the crown naturally, normal porosity everywhere else.

Currently using...
Poo: Any sulfate free or low sulfate poo
Rinse out: Tresemme Naturals
Leave-in: Camille Rose Moisture Milk
DC: Eden Bodyworks Coconut Shea DC
Stylers: Xtreme Wetline gel
Oils: EVCO
Moisturizer: Oyin Hair Dew/watered down CR Moisture Milk
Sealant: Oyin pomades or JC Nourish and Shine