I'll third that the Chained Oak is a great place.
Always had a nice stay there. You are able to use a different entrance to the park, ask Liz who runs the B&B which is easily walk-able, obviously saving you the parking charge too.
The views are lovely, and most of the rooms, barring the family suits, are located in converted barns and are more than adequate.
There is a lovely big conservatory off the house where you have your breakfast, this can also be used to eat evening meals etc in and the owners welcome you to order a takeaway and sit in there for the evening.
Whilst at the park you will hopefully have gone on the 'Hex' ride which is based on the legend of a chained oak. The actual tree the legend (which many of the locals still believe) is based on, is just a stones throw from the B&B, which is obviously named after it. If you ask for directions form the owners they will direct you to it, but there is a path leading down from the entrance of the B&B's driveway, if you follow that down about 100yds on your right, there it is. It's quite a sight even in its current state (a little graffiti & a few more branches fallen) its definitely worth a visit if you have been on 'Hex.' There are steps leading up to it from the path. But if you hear an old woman's whispers, RUN!
Recommend in the dark with torches.
EDIT: To add, you mentioned Public Transport, there is public transport available to the park itself, which you already know is so close.