I can't speak to the logistics of connecting your particular keyboard. But one piece of software that is useful for this (and has been around forever) is Scala. Depending on the abilities of your equipment, it can export tuning files in various formats or use MIDI pitch bends to allow real-time performance on an instrument that doesn't natively support such detailed tuning features.
And it can definitely do scales without octave equivalence as well as all sorts of other exotic tuning things. It does have a bit of a learning curve, and it's a little old-fashioned and clunky in terms of its interface. But it's probably one of the most flexible things out there for manipulating and building tunings.