I got a new display yesterday, a Newhaven NHD-0440AZ-FL-YBW 40x4 text display with backlight. Not a giant upgrade from the 40x3 display I've used, but this one can be bought new from Mouser, has 33% more lines, has backlight and also a HD44780-compatible controller. Soldering and changing the code to run took less than 4h, so not too bad.
During this I also remember why I have a secondary processor dealing with the keyboard and text-display: it takes A LOT of time to wait for each byte sent to the display. Running this on the Teensy would quickly eat up precious CPU-cycles better spent on heavy lifting (emulation).
I had missed the fact that the backlight draws 2W. It always pays to read the documentation thoroughly... Might have to fix some PWM control for the backlight level later.
Almost forgot: I could ditch the DC-DC converter used on the old display :D