Can I print my picture on {A,B,C}{0-10} paper?
Charcoal, 38 bytes
η⊞θ×⌊θ₂²Ｉ⊖Ｌ↨⌊Ｘ∕×φＸ²∕⁺²⌕ACB↥η⁸⁻⌈θ·²¦²¦²

Try it online! Link is to verbose version of code. Takes input as a pair of integers and a characters. Explanation:
η

Output the paper letter.
⊞θ×⌊θ₂²

Multiply the smaller dimension by √2 and use that if it is larger than the larger dimension.
Ｉ⊖Ｌ↨⌊Ｘ∕×φＸ²∕⁺²⌕ACB↥η⁸⁻⌈θ·²¦²¦²

Calculate the largest size of paper, divide by the larger dimension (less 0.2 due to rounding errors), square the result, convert to binary, and count the digits (because Charcoal has no logarithm function).