A303 is a bloody ridiculous road. Roundabouts every 1/2 mile. Single carriageway by stone henge and every bugger in a caravan wants to stop and have a look. It might be a shorter route, but its by no means a fast route.
Horses for courses though. Some family members are scared stiff of motorways, to the point they'll happily take 6 hours to drive from Epsom via the 303 instead of the m4 or m5 in 4 hours to wade bridge. Wadebridge from here in 3hours, yet Dartmouth which is nearer took over 4 hours via the 303.
There's another option following the m3 all the way down to Sputhampton and turn right through the new forest, but I've never been that way and google maps says its half hour longer than the 303.
My thoughts for you though, with a car full of pets. Keep the ship steady on the m4/5 and take a longer but quicker and more economical route with cruise set, and stop at the services for the pets for a
break brake.

303 is a good place to have an accident with all those roundabouts/side exits/dual carriageway slip roads that are too short for some people. I've rarely encountered such bad driving in one journey that on the 303. Avoid IMO.