I'm not sure about splits as I am struggling to work out a cheaper combination.
I see that you quoted the Route Via Salisbury fare. I take that to imply that you wish to travel via Southampton. Is that your intention?
Fairwater - Dorchester Stns
Route Via Salisbury
Off-Peak Return Restriction 8A
allows you to travel via Salisbury - Southampton Central - Poole - Dorchester South.
However please note that break of journey is prohibited on the outward portion, so technically you need to have travelled to Dorchester and back Poole in order to break your journey on the return portion.
... leaving as early as possible for minimum cost ...
If that is an issue, an SDR from Fairwater to Bristol TM is £12 with YP (allowing for the £12 minimum); the Bristol to Dorchester ticket can be used from 0700.
Will the £12 minimum fare be in place in July/August?