if i were to do such a project i would consider the following:
-connect the OLED to the axoloti, since there is nice support from @SmashedTransistors for those 1.3 inch OLEDs.
or, if the OLED has to stay on the arduino
-send a midi cc or program change to the arduino and let it display the names directly (you will have to save the names in the arduino sketch, which makes it static.
implementing it via sysex can be done of course, but it will be a bit of a pain i think. (formatting the text in axoloti to sysex format, send it over, convert it back to a string to then display on the arduino)
so the questions are really, do the names have to be dynamic, do they change? or can you just put them into the arduino sketch and select them via an incoming midi cc or program change?