I think you are right that the likely outcome of the meeting is "we'll consider this again and come back to you"
Then, assuming they do refuse, I've copied this from the website that bran gave the link for (another good one).
"If your request is refused, your employer must explain why. They must give a reason from the eight set out in the law, along with an explanation of why that reason applies.
You're also entitled to appeal. Your employer must hold a meeting to discuss the appeal within 14 days of you giving notice of your appeal. They must notify you of their decision within 14 days of the meeting."
Now I know that you are likely to have covered all the ground in your meeting anyway, but I do think you still need to launch an appeal (so that if you do end up at trubunal or wanting to get a settlement, you keep on putting yourself in the best light possible), but at the same time, if you can, I would slip in, either at the meeting or when you get the refusal, something like this. "You've been thinking about it really hard and unless there is something from the employer side that you are not aware of that impinges on their decision, then you do not think their response is reasonable or proportionate and therefore you have sought legal advice. Your solicitor thinks that you have a strong case, but has advised you to try to find an amicable solution and that is what you are doing, exhausting all the internal avenues first. However, they should be aware that you will pursue with your solicitor if their response doesn't change."
Not sure how big a company it is or how involved HR are, but in the larger companies I have worked for where line managers did their own thing until they got into trouble, it was usually when the employee said that they had talked to a solicitor that they called HR!!
ALso, at the meeting, I think you should be ready for them asking you what the consequence is of them refusing your request: ie are you prepared to resign and claim constructive dismissal or would you actually go back full-time, but find that you had lots of absences to care for your family or whatever, or you could just sigh and say mysteriously that you just don't know.
Good luck!
PS The underlying reason that I am banging on about you following the process is that IF you did decide to take it to tribunal, you have to create an application to have your case heard, and if they can see straight away that you haven't exhausted the internal channels, they may not be prepared to hear the case at all.