If you don't know already it would be good for you to find out if the school is allowed, legally, to diagnose a child with something like ADHD. It's not really appropriate to suggest it, though it does happen alot.
Just because a child is active doesn't mean they have ADHD. Gifted children are often very alert and need less sleep, my DD was like that. We did have her tested for ADHD before entering kindergarten because people would make comments to that effect, but the data wasn't there to support it. She's just very active when she's awake and learning.
The asynchrony is very hard but the gap slowly closes year to year.
It sounds like you could either try it and see how it works. What is their long-term plan with giving her extra work...is it just more worksheets of the same level that she'll have to do over and over? That would be very unfair. Is it more indepth things, or more advanced things? It sounds confusing for them to say they'll give her extra work but she has to stay with the curriculum, I'd make sure you understand the details!
If they suggested acceleration, what about subject acceleration where she goes to a different grade for that subject, if they can schedule it?
If you can afford it you could try to get her tested before she goes to school to have some data that will help you understand and advocate, or send her and see what they do and if she's fine just leave it alone for now.
It's not an easy decision!
