@SnowdayYay this is something I feel so passionate about I've just trogged up 2 flights of stairs to reply to you (I'm only logged in on my desktop).
Do it OP, get the assessment done. We were in the same position. Our DD despite being very bright was clearly having difficulty her brother and sister didn't have with reading etc. The school had limited resources to test an admitted the standard checks they did for dyslexia were a bit 'one size fits all'.
We paid for a private test - it took a couple of hours and cost about £450 from memory but it has been worth it's weight in gold. We got a detailed written report to pass on to the school (primary at that point) and instantly our DD got extra support, coloured paper and extra time in exams.
Because all this had been in place at primary DD's support was passed onto secondary where she got extra exam time, coloured paper in lessons, extra English support, a laptop, a reader and a scribe in some exams. She is now at college and the same support package is still in place as it will be if she goes to uni.
DD is just as smart as her siblings, it's just she has trouble processing information like words/letters and this support helps her achieve her true potential - She passed all her GCSE's with mainly 8s and 9s and the lowest was a 6 in English. Ironically whilst finding English touch, she is a wizz at maths.
Don't forget dyslexia isn't just about getting words mixed up, it can also be about processing and organisation skills. Our DD learns in a very different way to her siblings - i.e. not through reading/writing alone, but by practical discussion.
It has been invaluable and I would whole-heartedly recommend it. I really think if we hadn't paid for the assessment DD wold have struggled to get where she is today.
Please feel free to PM me if you want to know where we had the assessment done or anything else :)