Court days in summer: how do I stay polished without overheating or looking like a tech bro?

I work a public-facing job in California, not biglaw, and I spend a lot of time moving between courthouses and client meetings in the heat. I also commute by transit, and I already run hot, so this time of year I feel like a walking humidity cloud.

Everyone says layers, but I want a repeatable system that actually looks professional when I am standing at counsel table. The outer suit is fine; the problem is what I wear under it. Synthetics turn into a sauna, but some cotton tops wrinkle instantly and I look rumpled before 10 a.m.

What are your go-to strategies for looking crisp while staying cool? Specifically:

1. Favorite fabrics for shells that do not show sweat easily but also do not look shiny or cheap?

2. Do you prefer sleeveless shells, or lightweight short sleeves to avoid sweat stains under the blazer?

3. Any underlayer tricks (tank vs camisole, specific cuts) that keep the blazer from clinging?

4. Shoe recs for days when you might end up speed-walking across a courthouse campus?

I care a lot about user experience in other parts of life. I am a gamer and get irrationally mad at bad UI, and I want that same frictionless vibe in my work wardrobe: grab-and-go, no constant adjusting. Would love systems or specific pieces that have actually survived real court and office chaos.