Sorry, you're pretty much right. The equation I wrote is correct, but I naively plugged in a bad number: yes, P₀ V₀ = n R T = P₁ V₁, but I used the known number for V₁ (the final air in the container) as V₀. The proper formula is indeed P₁ V₁ ln(P₁ / P₀).

You have my apologies for my confusion.