I agree with the other mom. She probably isn't going to take a nap when there are other children in the home. My son finally learned around 6 months old how to put himself to sleep. At that time he couldn't sit up so I hung a toy in his crib that kept him busy and soon he would be asleep. He would take nearly 3 naps a day, then down to two at 9 months and the only time we held him and rocked him is with a bottle at night. This was our special time with him.
He's now 16 mo. old and he does cry some in protest when I lay him down for his one and only nap during the day but falls asleep within 10 min. He doesn't cry the whole time. Sometimes at night after rocking and giving him a bottle he may still not be asleep but we can lay him down with no protest from him.
It sounds as if your daughter is in such a routine of "hanging in there" with the other children you have in your home, her body has adjusted to being tired and still being able to go forward full steam.
If you start now, some how teaching her to nap during the day or, I know you said "no crying it out" teaching her to be self sufficient with putting herself to sleep at night you will only have to deal with it for a week or no more than two weeks. It will pay off if you are persistent. She is still young enough to learn. By the time they are one they change again and then again at a year and a half. Start now or you'll be stuck. If you want her to go to bed earlier so she will get up earlier between 7-8pm is a great time to start laying them down. My son is awake at 7am every morning and is in bed no later than 8/8:30pm. The only time it’s hard to get him to sleep when we know he is clearly tired is when he is “overly” tired. Then we have to let him lay in his crib and go to sleep on his own. He does fall asleep eventually and without too much fussing.
I believe you said she show signs of being sleepy even when she is fussing after you lay her down. My son has gone through stages in the middle of the night waking us up and when I go in there he probably had a bad dream. I picked him up, rocked him, he fell asleep, and I put him back down. I suddenly realized that he started doing this every night for nearly 4 nights in a row. He realized that by crying (which his sounds were different and clearly he didn't have a bad dream) I would come and get him out of bed and rock him. They learn very quickly how to manipulate and become attached to the bad routines.