How to select the perfect travel hat [closed]

I'm on the quest for a perfect travel hat. My requirements, in order of priority:

1. Sun protection. I'm balding. I don't like sunscreen.
2. Should not have an overly stiff brim that gets in the way taking portrait-orientation photographs.
3. It should travel well.
4. It shouldn't make me look like a doofus. I know "doofus" isn't a very precise term. And what looks good in one region may look doofusy(?) in another.

In the past I have tried:

• Baseball cap. Fails #2 and #4. And isn't great for #1, since the brim only keeps sun out of my eyes, and not off my neck/shoulders.

• A "fishing" hat. Works for #1-3, but fails #4. I don't feel self-conscious wearing this hat when I'm traveling the U.S. in a T-shirt and shorts, but in other parts of the world I prefer to dress a little nicer (long pants, button-down travel shirt), and then the hat looks really out of place.

Some research came upon the concept of a foldable panama hat, which really appealed to me because it's classy, but also casual enough to wear when I'm not at a formal event or whatever. The foldability makes it ideal for travel, where I don't want to wear a hat all the time (in a bus, on a plane, etc).

However, advice from Brent Black at The Panama Hat Company is not to fold/roll your panama hat:

...you can’t realistically expect any natural straw hat to look as nice after it’s been rolled as before.

People often compare fine Montecristi Panama hats with linen. Okay, let’s suppose you pick up your linen suit from the cleaners. You take your crisply pressed suit home, roll it up, and pack it in a suitcase. You jet off to your private island. You unpack. You and I both know that suit isn’t going to look like it did before it went into the suitcase. Same with a hat. And as with the suit, a little steam directed at the rumpled parts should spruce it up. (source)

So I'm left trying to find the perfect balance between portability, and fashion.

What should I look for in a hat that will travel well?

Or am I better off getting a nice hat, and just making sure I never crush it? Is this even practical when traveling light? (I will not be carrying a hatbox with me!)

Or perhaps the foldable Panama hat really is a good solution, and I should just accept that after rolling it a while, it just won't look as good--but will still probably look way better than a baseball cap or fishing hat.