If you need to avoid moisture, your best bet is to apply heat and shape the cap over a round object.
What comes to my mind is a bowl-shaped airtight food container that fits inside the cap. Fill it with boiling water, then seal the lid (please be careful not to burn yourself!), turn it upside down and pull the cap tightly over it. That should work like an iron, but from the inside.
The warm fabric is still prone to bending and creasing, so ideally you would let the cap sit over the bowl until it's cooled.
An alternative with (hopefully) less risk of burning yourself is to heat up dry uncooked rice in the oven at about 120°C / 250°F. Put the cap upside down into a bowl that has the desired shape and fill the hot rice in. The rice will warm the fabric and push it into the bowl shape. Let it sit until the rice cooled down. You have to be careful not to unintentionally cause new creases.
Maybe you'll find another alternative that suits you. The main goal is to gently heat the fabric like an iron would do and then let it cool in its desired shape.