I always assumed that TSMuxer produces an HD output exactly the same quality as the MKV. There's no quality degradation. Would I be right in saying this?

Yes. Tsmuxer is just a muxer -- nothing more than that. It puts the contents of your MKV into a BD (or AVCHD) structure. If the streams in your MKV are BD compliant, then you should do this. If they are not, then you have to re-encode if you want to burn a disc that will work in Blu-ray players.

Now for my main question: does an increased bitrate make a BD look better than the base MKV, or would there by no upgrade?

BDtoAVCHD is re-encoding your video. When you go from a lossy format to another lossy format, you always lose quality.

Really, BDtoAVCHD and tsmuxer do different jobs. They are not alternatives to each other. Unless you need an output that plays in stand-alone players, I suggest that you buy a decent media player and save yourself time and effort.