Equivalence of deterministic finite transducers - a special case of single-valued finite transducers - is decidable because it is decidable whether a transducer is single-valued. Note that two deterministic finite transducers $T_1$, $T_2$ are equivalent iff $T_1 \cup T_2$ is single-valued and their domains are equivalent which reduces to a DFA equivalence check on $T_1$, $T_2$ without output.
Can you point to a reference of an efficient algorithm to decide equivalence of deterministic finite transducers over finite words?
I am also interested in a decision procedure for infinite words assuming that all states are final. The latter transducer variant is also known as generalized sequential machine (GSM). Equivalence is also decidable for GSMs.