I have been to the Chase Sapphire Lounge by The Club, and it was very nice and roomy. I recommend giving it a try with your Private Pass card.
I did a review for this lounge, so if you are interested, you can see some pictures and info on my
Review of the Chase Sapphire Lounge
Keep in mind that since you only have a CSP card and not the CSR (Chase Sapphire Reserve) card, you can only visit one of the Sapphire Lounges per year, so consider that when deciding.