A concrete disc will do the job but like you said, the finish will be horrible. I would suggest using what's called a vanity blade, if you're in the US they might be called something different but is what they look like. You can use these dry, you'll want to do it outside and probably with some sort of overalls...
A dust extractor that will do wet stone dust is going to set you back a lot of money so best to just work outside for now.
This is the sort of router bit needed, you can get them with guide bearings too.