One other thing that would help would be if the A31 between the A338 and A35 were upgraded to fully separated dual carriageway. It is not that far and would make a big difference both for those going to Poole/Wimborne and to those further west, for the sake of 10 miles of upgradeed road you would have high quality dual carriageway from the M25 to Dorchester, which itself would relieve the A303 by making the A35 a viable trunk route as well as relieving the A37 and M5 as HGVs from Dorchester/Weymouth etc would divert A35/A31/M3/A34.
Not a lot of money in the big scheme of things but a big difference.
For the A303 if I was a politician with the power, I would persue bite size chunk strategy. I would dual the following as separate short schemes to make local improvements in this order:
Chicklade Bypass (Chicklade Bottom - A350) (D2)
Winterbourne Stoke Bypass (between Wylye bypass and A360 roundabout) (D2)
Monkton Bypass (D2)
Ham and Newtown Bypass (D2)
Newcott Bypass (D2)
Knightshayne improvement (to remove dangerous bend) (D2)
Ilchester - Sparkford improvement(D2)
A303/A37 roundabout grade separation,
A303-A3088 roundabout grade separation
Wylye - Stockton Wood Improvement (D2)
Ilminster Bypass Dualling South Petherton - A358 (to improve North - South connection) (D3)
A350 - Mere Improvement (D2+crawler lanes)
Horton Bypass Dualling (D2)
These would all be relatively short, cheap and mostly uncontroversial, and I would be emphasises that this would all be done for local reasons not as part of a scheme to upgrade the whole road, which is of course out of the question.
I would then use the monumental traffic jams that appeared on the A358, the mile or so past Stonehenge between the A360 and A344 and the half dozen or so sections of remaining S2 on the Blackdowns, none of which would be longer than 2 or 3 miles as a political battering ram to get the controversial bits, now reduced to very short distances, done.