Wear a bandana under your hat.
This way, you are abiding by your employer's wardrobe requirements while still being able to sop up your sweaty head.
One advantage of this technique is that you can swap out for a fresh bandana whenever you like. Store them in a refrigerator for a nice, but temporary, cooling effect.
I recommend folding the bandana in half diagonally and then fold it over itself several times until you have a rectangle with pointed ends. Then tie it around your forehead.
You can ask what types of hats are allowed, as some will conceal the bandana more than others.
If this technique does not conceal the bandana sufficiently, you can simply cut a bandana (or any piece of absorbent fabric) to the shape of the inside of your hat and fasten it (if needed) with a couple pieces of hook-and-loop fastener (or a couple safety pins). If you go this route, I recommend cutting a couple holes in it to improve its breathability.
Another solution is to approach this from a different perspective by exploring ways to keep cool.
In extreme heat while working on metal roofs, I've packed the inside of my hat with flexible ice packs. Of course, that didn't stop my boots from melting, but that's another story...
You can also purchase cooling vests that have replaceable cooling inserts. These are often used by people with illnesses that are exacerbated by heat (e.g. multiple sclerosis), but anyone can buy them.