pwd-hash like algorithm which can be computed on paper
What you're after is fundamentally impossible. Password hashing algorithms are inherently slow. They're designed to be slow even on a computer. Otherwise they would not serve their purpose, which is to make brute force search infeasible. A fast password hash is a broken password hash. You cannot possibly get anywhere near the requisite level of slowness with manual calculation.

For anyone with a computer, given a site-specific password that is calculated from a master password by hand, it will be trivial to go try billions of potential master passwords until they find a matching one.

And having potential multiple matches won't save you, because there can't be many of them, otherwise the number of possible site-specific passwords would be ridiculously low.