I have taken the kids to the ER with me when the need arouse. There is no need to call a social worker, that's a bit extreme don't you think? They understand that emergency's happen.
When you're sick it might be better to stop trying to put it off until you're able to go without her. Chances are you'll have an emergency during the night and have to get her up and take her with you. SO go ahead and go when you need to so you can lessen the chance you'll have to go during the nighttime hours.
I also say just take her with you. Do your best to keep her sleeping or at least sitting quietly. I usually lay the kiddo's on the bed and I sit in the chair. This way it shows I am giving the kiddo's priority and their rest is important to me.
If you do have chronic illness can you get a nurse to stay with you to help out? I had a friend who had some illness where she wore a urine bag all the time. She would get so ill sometimes that she had a written order for a nurse to come stay 24/7. This letter made the HUD office allow her a 3 bedroom apartment instead of a 2 bedroom for her and her daughter. She had to be able to supply that nurse with a private room to sleep in.
Several times a year a nurse would come stay with her due to her health.
It's just a thought though. If you have a true emergency and need immediate medical care then you have no choice except to take the kiddo's. They understand this and if they call child welfare on your for it then I would not trust those doc's to treat me anymore.
IF they put you in the hospital that's when you call someone to come get her. Just explain that you are being admitted and they won't let her come to ICU or where ever they put you and that you need help right away. That will show you have a support system and that you are doing okay.
Make a plan with several people you know. Tell them that you need to make a list of a couple of people that you can call at any time for help. That you will ONLY call them if you are being admitted and that you won't wear out your welcome with them. Then they need to agree to come, at any time, to the hospital when you are being admitted and pick up your child. This way you have a plan and a couple of backups.