Will I get good results using a flush trim router bit on 3mm carbon fiber plate?

Carbon fiber composite edges will not mangle like Kevlar. It cuts more like fiberglass but downward forces could cause delamination. If you can trap the edges between two layers of support material, you will prevent that. Something with clamping action would be best e.g. clamps or thru bolts. At work we do endmill and drilling using a backing board material. Standard carbide bits (only because they are cheap). Carbon composite eats bits, just like fiberglass. Diamond bits would last longer but cost a lot more.
Long sleeves and N95 will be fine. Carbon fibers are not as obnoxious as glass fibers in terms of poking into your skin. Suction would be really good. We use only house suction at work, with the vacuum system outside the building. That way particles finer than the bag mesh won’t get breathed in by the machinists.

@RocketGuy Thanks for taking the time to post your great answer! The sacrificial support material is a great idea. I’ll be sure to run the shop vac too.

Carbon composites is my thing. I used to work in the composites dept, with an office above the composites machine shop.