Deciding functionality of transducers over infinite words
Françoise Gire actually gives two proofs of her result. 

The first proof (I am quoting her paper now) uses a construction similar to the one used in [3] to prove the decidability of the $\omega$-equivalence of two functional finite transducers.

The second proof makes use of the technique you are describing in your question: $T$ is functional if and only if $T \circ T^{-1}$ is a restriction of the identity. The property for an infinitary rational relation of being a restriction of the identity is proved to be decidable in [3,4].

[3] K. Culik and J. Pachl, Equivalence problems for mappings on infinite strings, Inform. and Control 49 (1981) 52-63.

[4] K. Culik, Some decidable results about regular and pushdown translations, Inform. Process. Lett. 8 (1979) 5-8.