There are wooden silk screen aparatus that have survived decades under constant use. It is the standard. Metal is only shiny, it will rust, wood doesn't.
It's hard to know how delicate they are without seeing them, but I think you'll have to iron them.
This might sound crazy but as long as you're careful and patient it should be fine. Make sure the steam setting on your iron is off, it needs to be absolutely dry and that you have it on a low setting. Place a piece of brown paper over the silk to protect it from scorching - don't use newspaper or anything printed as the ink will transfer and it might be worth getting acid free brown paper which you can get from an arts and crafts shop.Then move the iron across it pressing firmly and working from top to bottom, be slow and systematic.
The other way to do, it if this sounds a little risky is to get some velcro, the stuff that sticks on rather than stitches. Apply a piece to each corner of the piece and then velcro it to a flat wooden board keeping the fabric taut. Then place another board on top and weigh it down with anything heavy you can find - big books bricks etc - it will take a lot longer and the results won't be as good but it's the slightly safer option.
Hope that helps.