Is it possible and safe to attach this type of seat to suspension seatpost?
Looks like the new one has the same kind of saddle rails so you can remove the attachment from it and try the bare saddle rails in the suspension seatpost. It should work!

Undo the two big black nuts and it should all fall off. 

Edit (Grigory, Criggie): The extra bits are just the clamp from an older design of seat post. So they’re not needed, but might be useful back on the seat post they came from. The common interface on the saddles is/are the two parallel rails.