It would be a help if you could give some indication of how much data you're talking about, i.e. number of bytes. Some devices (most) that may be able to store some data attached to a key will have a limit.
If however you can do something with the DAW, then this may well be able to 'send' a midi file containing midi and/or SysEx data, and this could in effect be unlimited. This would however move the possible problem, as you'd now need a way to create/maintain the data files, but there will be ways. You could adjust the system to have multiple data files, which might be sent in turn, in different combinations, to cover various circumstances.