If you're really running somewhere in the sub 10W range, you should be able to go approximately 5 days on a full charge theoretically. In Australia, I'd be fairly comfortable with that - even overcast days you'd probably get there. UK is a bit different of course

2-3 hours sounds about right based on those numbers (i.e. 240Wh).

In that case the extra battery may be OK since you do have that excess charging capacity. As long as typically the energy produced each day from your panels is greater than the amount you use you'll be OK.

How would you switch on the router? Can that controller turn on loads via its GPRS module? But can it then operate as a router? In that case wouldn't it make the router unnecessary? Or you'd need two SIMs/connections (1 x GPRS and 1 x 3G)?