OP I have found reading this thread really interesting and for what it's worth wanted to add in my DD's experience.
She is currrently Yr 6, working at expected in English, Maths and Reading. She's a late summer born too. and also not a prolific reader, and I have to say I think this is the main area where there was a 'gap' to be bridged when thinking about selective schools. For my DD we felt maths could be taught and grasped, but being a slow reader meant she struggles to build vocab, spelling skills etc
Where we live there is only one Grammar option, which this year reportedly had 3000 sitting for 180 places, and we knew this was never really in our DD's reach. I looked around a wide mix of state and private schools for Sept 23 entry, and ultimately, I came away feeling encouraged that the state comprehensive options around us were actually very good. The reason I mention this about comprehensives is that I think you have time to educate and familiarise yourself with the other non grammar options (of course you may already have done so).
We got DD a tutor in early Yr 5, with the view to sit various selective private school entrance exams. As time went on we realised that, mainly due to her not being an amazing reader, these private school exams would be too hard. In the end, she just sat one private school exam, which with the help of her tutor she actually seemed to enjoy. Wierdly, whether she passed or not has now ceased to be important to me. The tutoring has been great - its been a journey of helping her enjoy learning and the 1 to 1 has been amazing, but its also made us as her parents realise where her strengths and weaknesses are. It's ultimately a long game. Everyone develops at a different pace. 11+ may well be too early for our dd to really 'get it' academically but the process of working it out has just deepened our understanding of what our dd needs. This will be different for every child.
Sorry this is a long waffly post so I'll finish with this - Why not just get her the tutor, it will help and encourage her in more ways that just the end goal of Grammar. You'll feel and know that you are giving her a boost that it sounds like you feel she needs, an educational ally if you like! If she passes, great, if not it wasn't the right school for her.