Kinetics has made belt drive Brompton's.

But it seems more difficult for the Birdy because to fold the bike the chain must be on the smallest sprocket and belt drive sprockets are bigger than chain sprocket what could cause problems to fold the bike.

Another issue is folding the belt on the opposite side of the teeth what for many belts isn't allowed by the manufacturer.

For the Birdy suspension I have Multi-S suspensions (front and rear) the front Multi-S is real shock with a spring + an oil damper and adjustable spring pre-load.