Hi there
Again, a bit of a shot from the hip, but I've put together a map of things I think would be interesting, without being touristy:
https://maps.google.co.uk/maps/ms?msid= ... 4,0.583649
The thing with the forest is that the whole place is beautiful and generally speaking, full of ponies, so just about anywhere in the forest would be a good place to picnic and watch the animals.
The points I've highlighted are places where I would expect you to have better than average success at enjoying the forest, without the legions of tourists and cyclists that fill the place up.
If you're trying to avoid Beaulieu (which will consume your whole day, I think), then my recommendation would be arrange to explore with the car in the morning, have lunch at one of the establishments I've highlighted, and then have a picnic for your afternoon tea (maybe looking to eat at about 15:30 - 16:00). Although you won't be in Dorset, you're near enough to qualify for a Dorset Cream Tea, which is basically scones, jam and Dorset cream (and tea, one supposes) - these are nice to add to your picnic. Are you looking to stay overnight in the region, or were you looking to move to your next destination in the evening?
A side note whilst driving in the forest - there are two other users of the roads to be aware of:
Cyclists - you get 50 points if you hit ones wearing red - 20 points for all the other ones. Okay, just kidding. The point is, they're not especially aware in the forest and tend to do dumber things than they would in other parts of the country, so just be aware of them, as they still have right of way.
Horse riders. Please slow down when passing, as there are many horse trainers working with nervous horses who may not yet be used to cars. Also, sometimes, you will come across horse riders riding next to each other and generally taking up the road. They have right of way, so please be patient if you see this.
I'll update the map if theres anything else I think of. I'll check with the missus tonight.