I tried this tonight and it was slow enough that WAF was an issue. It's not terribly slow but it's different enough than just going straight to the Lutron bridge that she noticed.
All I did was set up one Pico to control one set of lights (all on same circuit / Lutron dimmer) using [RELEASE] Dimmer Button Controller (configure Pico to emulate Hue Dimmer or any button device to easily control lights).
I also tried the same test with the built-in Button Controller and the timing worked out the same.
In both cases, I found the lights start to change in about 1.8 seconds vs about 0.5 seconds using the native Lutron integration. That ~1 second is enough for my wife to notice...
I tried both fast and normal Pico drivers but didn't notice an appreciable difference either way. I realize some latency is to be expected but it didn't sound like it should be very perceptible. Am I expecting too much or is there something I should be investigating that would make this quicker.
If it's about as fast as it'll get then I'll need to investigate using the Picos in other locations but will need to leave these as-is I think unless there's a way to keep the native and "add" behavior on top of it.
Edited to add that i just tried using the button controller on top of native Lutron controls and it actually seems to behave like I’d want. It’s quick to respond and on gets limited by what I set in HE for the first press level.