Ahhh!! We flew BMI in April and they told us it was first come first serve with a car seat and we couldn't book in advance - which is totally twattish but there you go.
We decided to take our own car seat and made sure it fitted with their requirements and was the make they use anyway - Britax. We get there and they REFUSE to take the car seat - we had to wrap it up and it went in the hold and they also had no car seat for us to use. In anycase, they told us that DS - who was only one month away from two, had to go on our laps for landing/take off anyway. They are full of conflicting information - web says one thing, phone says another, check in desk says a third thing.
My advice - if you are going to be using the car seat abroad, then wrap it up really well - bubble wrap etc with the instructions to hand so they can see the make/dimensions without opening it and then if they say no, you can't use it on the plane (which most likely they will!) its all packaged up and you don't have to run around the lounge trying to find bubble wrap and tape!!
In any case, after take off, DS was more than happy in his own seat and we could see an interact with him a lot easier so it was better than having a car seat. We also took a portable DVD so that helped a lot to pass the time for him
Hope this helps? Take some grub with you or feed her before she gets on the plane - I found BMI child food absolutely shite and no difference to the adult food - they gave DS a cup and saucer!!! No kiddy juice box etc!