Language to Construct Finite State Transducer
Yes, such a specification would work. Basically a FST is a finite state automaton, with different labels on the edges. We can go from FST to regex by a standard algorithm. The same method works with transducers, only the labels are now pairs of strings. The concatenation of labels is done component-wise: $(a,x)(b,y) = (ab,xy)$.

There are several types of FST. To obtain a general one both input and output component are strings, not just letters. It is not difficult to see that alternatively one might require that both input and output are either a letter or the empty string.

All said, I personally prefer as a "language" the finite state graphs: they are easy to "program".