I’d wear a casual dress & sandals. Something like this (without the jacket) or this. If you feel that’s TOO casual, it’s easy enough to go slightly more formal with a dress.
Bonus: it’s like wearing shorts, comfort-wise, but people think it’s dressier!
Somehow, I don’t think a guy in a dress fits “Smart Casual”.
Or just a cashier, for men, too. I stopped wearing polo shirts in general when I was constantly being asked if I “worked here” in just about every store I went into. I would say a long sleeve buttoned shirt for men but I really don’t know what to recommend for women.
Oops, I read the post about the OP being a woman, but not the followup post where elbows clarified she was not looking for advice for herself, but rather a man.
This might work - I think it looks great!
OK - a tropical locale -
An expensive “Hawaiian” style shirt (Tori Richard, or Tommy Bahama, with a not-loud pattern), or linen shirt. They should be easy to find.
Linen trousers.
Weave pattern slip on shoes.
If you want to look good you have to pay for it.
Hmm, this seemed a bit interesting, but searching around for collared tee-shirts brought up either regular t-shirts with fancier stitching around the neck hole) or just plain ol’ polo shirts (a style I know well, especially pocket polos). Can you flesh out this collar t-shirt idea a bit more, perhaps it could be ground-breaking.
Of course, if the t-shirt has a dog face on itthen it’s always stylish… 
For the OP, put another vote for long sleeved button down shirt, slacks, and dress shoes (not sneakers) as the minimal default male “smart causal” attire, and take it from there.
My Daddy us turning in his grave. He was a clothes horse. No T-shirts ever. To him that would be like wearing your underwear. I personally like a nice T-shirt under a sport coat. For a tropical climate, pick light colors and light weight. Nice chino pants.
Which tropics also matter. As I understand it, in say Key West, “smart casual” is literally no flip flops, no bare feet, (maybe) no shorts. Clean pressed T-shirt would be overdressed. San Diego would be different.
In the Caribbean or Central America a guayabera is acceptable almost anywhere.
This is the correct answer. Absolutely not a t-shirt and neither of the collar short sleeve polos, either. Wear a button down and nice pants and jacket. Better to be a little overdressed than under. You can always take the jacket off and roll up the sleeves for a more casual look.
This works as “smart casual.”
That’s not always true. She could also work at a summer camp or on The Love Boat.
Sorry, but in a tropical climate that would telegraph “I’m not from around here.”
The simple rule that I have always seen is: T-shirts, no matter how decorated, are never appropriate wear for any time clothes matter. If someone goes to the trouble of specifying clothing rules that by definition leaves T-shirts out.
T-shirts are fine for exercise or athletics or physical work. Not for social occasions.
And frankly, Mr. Downey didn’t pull it off. Came close I admit, but T-shirts aren’t appropriate for sport coats. And I say this as a man with a drawer full of T-shirts that i wear all the time. For exercise or working in the yard.
I’m the furthest thing from a clotheshorse, but that photo screams dork to me.
To be honest, anyone specifying a dress code for an invite to their house is an arsehole. If you can’t trust the people you know to dress themselves in a presentable way then best not to invite them in the first place.
If the group is going to a hotel afterwards that has its own dress code then they can make that known but seeing as jazz is mentioned and the host seems fixated with appearances and specifying a “fine” meal, “fancy” hotels and dress codes I’d be crying off with a migraine.
By that definition, what makes something a t-shirt? Because there’s a lot of variation in knit pullovers.
I’d say the lack of a collar is one clue that you’re wearing a t-shirt.
