For most 5-point harnesses, if she is rear-facing, the straps should be at or below the shoulders. If forward facing, at or above the shoulders. The tips of her ears should also be lower than the top of the shell.
Some harnessed seats have low size limits. At 3 years old, she is too young for a booster so you will have to get another harnessing seat with a larger height/weight capacity.
Look here to find local help: http://www.safekids.org/in-your-area/ You can search for Child Passenger Safety Techs and upcoming inspection events.
People seem concerned about the coat issue, so I'll offer this: to test the coat, put the in the carseat with it on. Then without loosening the straps, take her out. Remove the coat and put her back in the seat. If you have to tighten the straps to get them snug again, the coat shouldn't be worn in the seat. If that is the case, you can put her in the seat without the coat, then put the coat on her backwards over the straps. Hood down, of course. :-D
I have a 12 year old in a booster, because he does not fit in the seatbelt. It cuts him right across the stomach and neck. Even though our state law only requires it to age 8, that is clearly not the appropriate choice for him. If you treat car safety in an unemotional and factual way, kids won't even think about being embarrassed.