If you're 'not going to keep any of the furniture', why would you even buy it furnished?
Proudspire has a lot of the Greymoor paintings, which can drop from Treasure Chests and be bought from traders, but are currently the most expensive furniture in that house. Proudspire also has a few of the Purple Solitude armchairs and other pieces, which also cost relatively more than other styles, but there aren't that many of them as I'd say around 50% of the house is furnished with the 'Rustic' Solitude furniture. (For example I found the prospect of getting so many Bound cheapo desks/benches/crates for my Crows off-putting so I bought it Unfurnished with gold).
Even if you would want to keep some of the Solitude paintings/furniture for other houses, it might be cheaper to buy them with gold, depending on your platform. If you're on PC, browsing guild trader item listings with AddOns will make it easy to find what you need; on consoles not so much.
It also depends on your Crown/gold trading stance and funds. If you know that you could get 2 million gold for your 7k Crowns in your trading discord for example, and you know that Solitude items are 10-20k each at traders, you can do the math whether it's more worth to you to use your Crowns on furniture, or change it into gold and use that to buy the house and the furniture.