Thanks for the quick reply 
Full duplex it is then, I guess! The price is about the same but my favorite shop only has the half-duplex version. But I can get the full-duplex ones elsewhere.
The amount of data being sent back and forth is not a whole lot. The PC sends a request which is at most 20 bytes and the Arduino replies to that, the replies' lengths depend on what kind of request was made and will be anywhere from 1 to maybe around 30 bytes.
To eliminate the need for high-speed communication the Arduino does most of the processing. Basically the Arduino does most of the hard work and just sends an event log to the PC so it know's whats going on. A high estimate would be 100 bytes/sec from Arduino->PC and 200 bytes/sec from PC->Arduino (and much less if I move all the scoreboard code to the Arduino).
So even 115kbps is way more than I need.