Thanks Grabby. It's a little rigid; this seems a better overview to me: https://www.webmd.com/children/what-is-hyperlexia
Hyperlexia is not, for the most part, an official diagnosis that's given out, and the criteria can depend where you're looking. It's also not specifically used in the ASD diagnosis criteria (as you presumably already know given your experience). I mentioned it because something that could be described as hyperlexia is common enough among autistic children (compared to non-autistic children) that "early/self-taught reader" always makes my ears prick up, if there's any other reason to be thinking along the lines of autism. But it's not always as cut and dried as the lists online.
Using myself as an example of what few people would dispute as hyperlexia:
I learnt to read without instruction, maybe from being able to see the page while my parents were reading me stories. I've been told I began recognising symbols (like sweetie brands, maybe) at 18m, by 2 I would read out the front of each food packet as my mum put shopping in the trolley, and by my 3rd birthday, I could apparently read pretty much any text fluently and accurately (with occasional mispronunciations of unfamiliar words with ambiguous spelling, or sometimes misplaced emphasis in long sentences), to myself or out loud. My understanding would be that of a typical 3yo, of course — autistic hyperlexia isn't necessarily an intelligence thing, more a quirk combined with a passion 
Obviously some children who read early do so because they're just very intelligent, not because they're autistic.
At infant school, they assessed reading ages and I came out at 14+, i.e. adult, on the test they used. And at pretty much at all times when I wasn't actively required to do something else, I'd be reading, right through to early adolescence. I even think in a kind of text ticker-tape most of the time.
I guess that's a fairly extreme version; it doesn't have to be quite that early to count as hyperlexia, just earlier than expected — it can describe someone with LD who might not be expected to learn to read at the usual time who nevertheless does, for example.
I know these things about myself mostly because of all the forms my mum had to fill in when I went for an NHS assessment for ASD, which I was then diagnosed with. There were (many) other signs in my childhood, but they're not ones that necessarily appear on some of the hyperlexia lists — not responding to being addressed but having no hearing problem when tested; serious behavioural problems especially at school; social problems at school; extreme sensory sensitivities and restricted diet; speaking like a little adult or in full sentences; eating paper, a lot, for years and years, despite getting into trouble repeatedly for ruining books (I never ate the bits with words! ); being described by adults as "tactless" or "gullible" or "different"; etc. etc. etc.
My speech development was entirely un-delayed, though, with an advanced vocabulary (all that bloody reading…), no echolalia, and no difficulty understanding spoken language or initiating conversations. My eye contact was fine. By that list I wouldn't qualify as hyperlexic. But there are so many people like me — with this type of early and obsessive reading and an ASD diagnosis, but who weren't overtly impaired in verbal communication — that I think the construct needs to be rethought a little, to include subtler signs of communication difficulty in particular, and ASD in general.
The hyperlexia construct is still very immature, essentially, and should really just be seen as suggestive at most, IMO. Early reading doesn't necessarily mean hyperlexia and hyperlexia doesn't necessarily mean autism. But they do go together enough that, like I said, hearing "early reader" when there's a possibility of autism (like maybe there are other signs, or it's in the family) does get my attention — the WebMD article I linked has some interesting but seemingly unsourced stats.
I'm sorry, I've gone way off on one, just in case any of this is interesting/useful/relevant to you. It's entirely possible that your DD is just an intelligent little girl who therefore learnt to read a little earlier than some other children, and completely unrelatedly happens to have an autistic sibling and to be struggling with a tough patch of behaviour and maybe a double standard in expectations. And that sentence wasn't meant to sound sarcastic; I genuinely think it's possible, maybe even likely. I just wanted you to be aware of the fact that sometimes there can be a link between autism and early reading, and it doesn't always fit the rigid pattern of the checklists.