Now, I can't read German, but someone suggested using Sugar Starch to stiffen the fabric. I googled the recipe, stole a balloon from my teaching resources for my childbirth classes (yes, really I use balloons and ping pong balls in my classes) and made the syrup.
Sugar Starch recipe
1/4 cup water
3/4 cup granulated sugar
Mix water and sugar in a small pan. Stir the mixture over low heat (do not boil) until clear and not sugary. Remove pan from heat, and let mixture cool.
Once the mixture had cooled down enough, I dipped the doily in it and smoothed it over a partially inflated balloon placed upside down in a noodle bowl. I really mooshed heaps of the mixture over, which is a bit messy and takes longer to dry but I wanted to make sure it would really get stiff.
I left it to dry overnight. In the morning when it was harder, I popped the balloon by sticking a couple of layers of sticking tape on the balloon then poking a couple of holes in the balloon through the tape. This stops the balloon popping suddenly and especially good when you have delicate media you are working with.
I was left with a mostly dry product, which I then left to dry near the window to fully harden.
Once hardened, I used a clear glass candle holder in it and a tea-light candle. I did this as I wasn't too confident that the direct heat of the wax in the candle wouldn't soften the starch. If it did, I guess the beauty of using the sugar starch is that you wash the doily and start again.
I have seen tutorials where PVA or clear-drying craft glue is used in the same way, and I think that would be a better option for more permanent projects!