You know what? I know this taps into stereotypes, but I'd rather people with little to no money visit Chatsworth House, go to the library, go to the theatre, access sports and healthy diets etc. It is good for us all. Why are you so invested in wanting people to sit in their house, drink cheap beer, eat UPFs and watch tv as that's all they can afford? Why do their children need to be deprived because their parents struggle financially? Why is it that money needs to buy access to art, beauty and education?
We are lucky in the UK that you can visit many world class museums for free. If you want something for nothing, there is actually plenty available. You may know that Chatsworth is private property, i.e. not owned or run by the taxpayer, and British nobility has, in fact, a long history of giving to the disadvantaged.
People on benefits should have the opportunity to further themselves and develop, this is cultural capital. Did you know they have access to some educational courses on free/ preferential rates also?
I am tired of living in a society where it's a permanent race to the bottom, rewarding the cheapest, crappiest things. Learning about arts, history and landscaping shouldn't be for the privileged. I don't want to live in a world where about 5% of the population can follow a conversation about arts, politics or history, because the rest think education is 'not for the likes of us'. Unfortunately we are largely there.
You may well have 12x more money than people on benefits. Say after all essential expenses, someone on benefits may have £100 spare. If you have £1,200 available to spend, including whatever you choose to save, you have 12x more money. If you have £600, you probably still have 12x more money.
Regarding specifically advice for your visit. You can park for free in a nearby village and roam the grounds for free. You're welcome.
And here's 50% off if interested: https://www.artfund.org/explore/museums-and-galleries/chatsworth