Quick answer - NO.
The Christchurch and Bournemouth area is (like so many other areas these days) pretty poor. I.m struggling to know of any WG's .
An AW search of BH23 (the Christchurch post code) only brings up 3. BH25 (New Milton - about 4 miles away) shows 3 and BH24 (Ringwood - about 6 miles away) shows 2.
Although you appear not to want to go into Bournemouth, its your best bet for a punt.