I would like to name two paths forward: “intonalistic serialism” and “intonalistic pitch class sets”.
For the first, I can imagine a new serialism where a) there are more than 12 notes, b) certain intervals are allowed (those that can be tuned) and others disallowed (those that can’t be tuned). The question of the fifth to minor seventh (untunable as used in traditional dominant-tonic harmony) is left for later.
For the second, it would be great to develop some of the tools used in manipulation 12-member pitch class sets (and sets with fewer members) with regard to the tuning of the individual elements, and perhaps extend to sets of greater numbers. As a first step, perhaps, the 8 note diatonic set of tonic, low supertonic, high supertonic, mediant, subdominant, dominant, submediant, leading tone. Then the 10 note set of tonic-dominant harmony. usw.