Can BPS be used to encrypt a single integer, and restrict the output to a certain range?
This can't be done in a straightforward way with just BPS, since the straightforward way will result in a year greater than the current one for roughly 80% of ciphertexts (since 9999 is the biggest four-digit ciphertext possible). However, using what's called a "rank-then-encipher" scheme, you can specify an input and output format using a regex, then build an FPE scheme on just the strings matching that regex. See this site for details.

EDIT: Also, I don't know BPS that well but I think the "integer of a certain cardinality" can just be a base-10 digit, so you can encrypt a four-digit year as four characters.