I'm not sure I'd even bother submitting a sun path diagram (which will no doubt cost you): the plans have to show the north point anyway, so unless it's maginal it should be clear enough. A simple satement about where shadows will be cast should be sufficient I would have thought.
You have a right not to be overshadowed by a new development, but this isn't absolute (ask the people round Landsdowne!): it all depends on the context and degree of overshadowing. On the other hand, you have no right to maintain an existing view, nor to not having your property devalued.
I know it's hard when it's your own home that's involved, but try and keep matters as objective and businesslike as possible - the emotional cost can be very high if you take it too personally. If you're still on speaking terms with your neighbour, I'd let them know that you see what they're saying, you've done your best to accomodate their views, you're quite happy for them to put an objection in (bite your tongue on that one!), and that you think it's best for an independent 3rd party to decide, and hopfully they'll agree.
The council will decide as an independent 3rd party: unless you're doing something completely unacceptable, it's likely you'll get permission. There may be conditions attached, though again, they're unlikely to be major ones. One thing is that the council will only look at valid observations (i.e. on true planning issues): they're quite capable of ignoring spurious points.
From what you're saying, I would be prepared to have the approval appealed if you get it. This is a real pain, as it delays matters quite a bit, but again I'd try not to worry.
Best of luck!