I actually called the migration office in Neuchatel and this was their reply:
In principle, a student must stay in the canton where he is studying. Exceptions are possible and are examined on a case-by-case basis, but the travel time should not be too high.
So based on the facts stated above I think I have a high chance of approval.
I’d just explain the BeNeFri agreement in a letter and get it signed by the responsible of the Master’s program and annex to the permit application. Even better if the responsible of the Master tells 2 courses in Bern, 1 in UNINE and such details.
Student permits are slightly different, especially for non-EU. You are generally required to live in the canton where you are studying.
Although the OP says that he will be doing courses at three universities I guess he is actually registered as a student in Neuchâtel so should technically reside there so he will have to apply for the exception.
Indeed, so is it recommended to clarify why I need to reside in Bern but am registered in Neuchatel uni during the student visa application or after I come to CH and want to convert my visa D to a permit?
Yes, write a letter and explain the situation. Reference the BeNeFri agreement and see what they say. Worst case, move to Neuchatel, get permit and request to move to Bern because your doing more courses there etc.