I know this is going to draw some flames but....
At about $12 apiece you can't beat a Polo (as in Ralph Lauren) sleeveless. They are long and they don't shrink. I've had to recycle them as cleaning rags (gun cleaning of course) due to wear holes after a few years and a few hundred washings...can't beat that for value.
I worked in Houston for several years in the days before "casual days" and I can tell you a light wool suit is steamy there. I can't imagine a dress shirt under a suit without a light sleeveless tee shirt to catch the sweat and aid preventing getting raw. Also, I'd recommend linen as opposed to cotton casual shirts for the climate there.
My summer dress (now that I'm retired) is a linen shirt over a sleeveless tee with the tail out on the shirt (I like the guayaberas). They conceal great and the tee cushions the IWB or shoulder holster.