No, it won't work.

The rear light output of all front dynamo light excepted Supernova is AC voltage as coming from the dynamo hub while the rear light output of the Supernova front light and input of the Supernova rear lights is DC voltage.

For the Shimano hub, only the new Mircospline accept 10t smallest cog, no 9t cod and as already said the cassette with the smallest biggest cog has a 45t biggest cog too big for the Birdy (the chain rub against the right rear arm). 11t smallest cog gives a too short longest gear inch with the 52t front chainring and would require a huge chainring with about 63t to have the same gear inch as 52x9.

Its the reason to choose XD or XDR for which there are cassettes with 9t and 10t smallest cog and not too big biggest cog. Campagnolo Ekar has also this type of cassettes but requires a very rare Campagnolo n3w freewheel