I would certainly recommend returning this nbb (no back booster) and getting a harnessed seat for your daughter as this is NOT a safe option for her at 3 yrs. old.
Some people here have their 3 yr. old rear facing, but a properly installed and used front facing harnessed seat is perfectly safe too
If you posted her torso height that might help to recommend seats too. You sit her on the floor cross legged and measure from her bum to shoulders.
What is your price range?
Some seats you might like are the Graco Nautilus apx. $140-$180 I think and it goes from harness to hbb (high back booster) to nbb. It's a great seat. A more expensive seat would be the Britax Frontier 85. It has some more extra features than the Nautilus, but doesn't turn into a nbb. It is significantly more $ (I think $280ish).
Also, you could look at the Evenflo Maestro as a more economical choice ($80 I think).
Hopefully that gives you a start and then some of the techs can chime in with some advice too :thumbsup:
Really, the concern about a booster at her age is multifaceted:
1. Her bones are not ossified to be able to support her in a booster
2. She won't fit properly (as you said the seat belt is up in her neck)
3. I think she would benefit from the side impact protection of a harnessed seat
4. She is most likely not mature enough to be boostered until age 5-6ish because she won't be able to sit still in perfect position for car rides
5. I am sure there are more reasons too
Does this help you? Welcome, and I think it's great you came here to find the best seat for your daughter :thumbsup:
Ooops, cross posted a lot of people - and here I thought I was being good trying to help . . .