I have been a type 1 diabetic for 30 years. (I was born in 1957.) And my condition is exceptionally well controlled.
I have made a number of posting on this Forum relating to how I manage to achieve such control - and how I manage, safely, to get such low HbA1c readings. But your enquiry concerns something much simpler.
Having injections has never bothered me in the least. But one thing I will never do is to use pen-injectors.
I find them big, heavy, cumbersome and clunky.
Instead, I use disopsable syringes. They're simple, small and neat.
Okay, they still have a needle at the end. But I believe that if I were a 2-year-old, I would find them a lot less threatening than the great big gadget which (almost certainly) you are now using to give your child her injections.
The pen-injector you probably use is, I would guess, not much smaller than your child's arm.
The syringes I use are tiny: 0.3 ml. They are the size of my little finger.
There may be a slight problem if your child's insulin only comes in the form of a pen-injector vial. (One of the insulins I use, Novorapid, only comes in such a form.) But this problem can very easily be got around. Please ask me for details.
I might add that I never use those aggressive spring-loaded gadgets to obtain a blood-sample, either. (I'm sure that small children would find the clunking noise they make quite frightening.)
Instead, I simply use a bare lancet to gently jab the side of my finger. They are so exquisitely sharp, they're quite painless.