There are indeed few tires available in ETRTO 355.

Besides the Schwalbe, there are also Panaracer Minits Lite PT in ETRTO 355 but they are very difficult to find outside Japan.

There are also some BMX tires in ETRTO 355 most of there are very wide tires (more than 50mm wide and won't fit on the Birdy I think).

I changed my wheels for ETRTO 349 (Brompton wheel size) that is only 6mm smaller because there are now much more tires available in ETRTO 349 than in ETRTO 355.