My son is also in first grade. This is our school's first year using Common Core and he hasn't had any sort of mid-year test. I haven't heard of one coming, but can't say for sure that it won't. They have had a few math tests - mostly unit tests - and I feel like the classroom work adequately prepares him for those.
I think you are wrong to assume that your daughter won't do well on the upcoming test. If she's keeping up with the work on a daily basis and her teacher feels she's on target, there is no reason to expect the test to be too hard for her. I know that you've seen it, but maybe she is more capable of passing than you realize.
My son's teacher really stresses comprehension. When we read at night - whether I read to him or he reads to me - we ask each other questions about the book. I ask him a couple and he asks me one or two others. By asking and answering questions, we are preparing for the type of test that you are describing. If you aren't doing this at home, make sure you start. Your daughter should be reading out loud to you for at least 10-15 minutes every day, in addition to the time that you spend reading to her (ideally, at least another 10-15 mins).
Don't stress about it before it even happens. Your daughter will pick up on the stress, feel more pressure, and likely do worse because she's so worried about it. Nothing you said gives me a reason to think she will fail. I do not believe that turning 6 in September makes her too young for first grade. Several children in my son's class have September and October birthdays and do just fine; others have January birthdays (so turning 7) and struggle to keep up.
You are jumping the gun in deciding whether or not she's ready for second grade. You haven't even seen the test results yet. She may pass with flying colors. YOU are the only one who thinks she isn't doing well on the assessments - her teacher has repeatedly told you she isn't concerned and that your daughter is progressing just fine. Stop using her age as an excuse to fuel your worry and question your decision. She is six years old, which is the exact right age for a first grader, and she is keeping up with all the work.