I think there can be an issue where a school is not accurately assessing a DC's level. Over time I've read a number of threads (here and on TES) where both parents and teachers discuss the issue of keeping levels lower than they are because of the pressure to show the correct amount of progress.
A child confidently reading and understanding chapter books in year 1 may well be working within level 3. What would the expected progress be for the year 6 SATS for this child? Certainly a level 5 and depending on if they are a level 3b or 3a possibly a level 6. English level 6s are notoriously hard to achieve, the difference of level 6s achieved nationally in maths v English is quite marked.
FWIW I don't agree with this pressure to show a constant and linear amount of progress, although I do agree with assessing children accurately so that teaching is geared towards the current learning needs / appropriate next steps.
Bornfree have you considered downloading some old SATS papers? It depends very much on your DD's personality of course. I did it, although for completely the opposite reason. DD's teacher assessed her as working comfortably within level 3 and I didn't believe the teacher 
. I downloaded an old SAT paper and DD's score left no doubt in my mind - however DD is very happy doing workbook / papers etc. And before I get flamed when I say very happy I mean "this is great" & / "I love this mum type comments 
.
This would provide you with some hard evidence because whilst a SATS paper is just a 'snapshot' and not the same as working consistently within a level, it is still indicative, with the responses written being quite telling of ability and understanding. Start with the level 2 paper and if she comfortably achieves a 2a then you could try the level 3 (assuming she's happy to do so).
Just for comparison my DD is also year 1, has coincidentally just finished Matilda too and is on level 13 at her school (so 2 levels above lime). She gets a range of texts from school ranging from complex non-fiction to 6 chapter fiction books (some are scheme, some are off scheme), they are interesting and challenging to just the right level.
If your DD's school have her assessed well below her actual level then they will certainly not be challenging her within school, nor differentiating her work appropriately. She's there for most of the day, whereas you have her for a limited time only. Why wouldn't you want her assessed correctly?