The galleries are worth a visit, but won't take too long. If the weather is cool could you leave the dog in the car for half an hour while you look round the galleries first? The grounds are massive and you can walk a long way. They don't like you eating picnics outside the picnic area, probably to avoid litter.
The vast majority is outside. There is a visitor centre with a gallery but it is just a small part of the Park. There are some lovely walks with plenty of sculptures to see on the way. We were there for the whole afternoon and didn't see everything. You can pick up a trail guide when you arrive - perhaps it's available online so you can check out exactly what's available to you before you go?
Thinking of going at half term and they say they're dog friendly but no dogs in buildings, galleries, upper lake and menagerie wood.
I've not been before so not sure how much we'd miss out if we couldn't see these bits. Would it matter or is there plenty more to see? Are the galleries very big? Dd and I do like art but maybe we could go in separately and swap dog holding duties? But not if they're big galleries.