I don't have a practical bike but if I may, I will make a small route suggestion which you might consider since you've said it's not set in stone.
After Haslemere, if you head Eastwards towards Selborne, and then South towards Froxfield (i.e. routing north of the A3 rather than south as you have it now) there is an absolutely fantastic descent into Petersfield from Froxfield. As far as I can tell that road is unnamed but you can see it quite clearly as it winds it's way through Ashford Hangers Reserve on googlemaps.
It might not be convenient but it would certainly be fun. Selborne and Froxfield are also very pretty. I was on my touring bike there last year and it was lovely.
I don't have a practical bike but if I may, I will make a small route suggestion which you might consider since you've said it's not set in stone.
After Haslemere, if you head Eastwards towards Selborne, and then South towards Froxfield (i.e. routing north of the A3 rather than south as you have it now) there is an absolutely fantastic descent into Petersfield from Froxfield. As far as I can tell that road is unnamed but you can see it quite clearly as it winds it's way through Ashford Hangers Reserve on googlemaps.
It might not be convenient but it would certainly be fun. Selborne and Froxfield are also very pretty. I was on my touring bike there last year and it was lovely.