Just another thought. I'm not sure you big/small your 1 year old is. But, I assume since you are talking about bases then your child is still in a bucket type seat. This is usually the age that most kids grow out of the bucket and into a regular 5 point harness car seat. If the childs head is 1 inch from the top or feet hanging over the edge it is time to change. Rear facing bucket with feet hanging over into the back of an airline seat will not be comfortable.

If you are at that time now, you may just want to get the next seat for him/her and bring that with you. No need to worry about the base anymore.

We flew with our DD a lot. She always had her own paid seat and once she was old enough not to bring the car seat on the plane, we checked it with baggage and used the "cares" airplane seat harness for her.