It has to be said, but Chatsworth is well worth a visit if you’ve never been. We live locally and still visit 3 or 4 times a year.
You don’t need to go into the house or gardens, there are loads of walks in the grounds. The one we like most is to walk past the kids adventure play area and up the hill side to the hunting tower and the Emperor Lake and surrounding woodland.
The main part of Chatsworth is busy but as is the norm people won’t walk more than 250yds from their car so get up the hillside and the people thin out rapidly.
You can walk over from Bakewell too over Calton Pastures if you fancy a longer walk.
Due to the area there are loads of websites with White Peak walks, and most cafes and pubs are dog friendly.
What village are you in – we could be bit more specific then.