You should address your concerns to the appropriate representative body and ask them to raise them with the faculty.
You need to establish clearly if this pattern of behavior is typical or atypical for people with similar responsibilities in the institute. Ask other groups for their experience. If at all possible try to get some stats together to support your case.
If there is no (student ?) representation body that can deal with this then, as a group you need to formally contact the head of the department and assert the need for this issue to be addressed without delay.
It's for them to work out a strategy to do that.
The initial suggestion will (at a guess) be that you "talk to the professor". This of course is just a way to avoid responsibility. You have no authority to negotiate with this individual and they seem not to be able, for whatever reasons, to handle their workload. It's the department which needs to shoulder this responsibility.
If the department refuses to act then you may need (again ideally via a representation group) to address this matter to a higher authority within the institute.
No, it's not reasonable for this to consistently happen. The purpose of that institute is to teach and research. Mentoring is not an optional extra, it's a core function. The issues here are simple :
- Meetings cancelled at last minute because other last minute things crop up
This (if true) is departmental chaos in operation and while you might expect the occasional meeting to be cancelled for these reasons, they're not running a crisis management center for the police, they're running (or not running well) an institution where student requirements ought to be an integral part of the working day. There should be time allotted to this and that allocation should be respected. Exceptions happen, consistent issues should not.
- Meetings cancelled for personal reasons
Well this happens, but if it's happening a lot then, again, it's a departmental issue to properly cover the work. It's quite unfair to students (and others) to have someone whose personal life consistently interferes with core working responsibilities. Again it's a case of occasional being normal and a high frequency of these issues being a real problem that needs a real response from the department.
- But there's rarely any follow-up for makeup meetings; if there is, it never moves past us giving our research professor our alternative meeting availabilities.
This is another sign of either departmental chaos or individual chaos. There's a clear issue here that needs to be addressed. When you were taken on it was to mentor your development. If this is consistently not being done properly then it's a departmental level failure (at least).
Very, very important.
Lastly in fairness to the individual you need to note that they may be severely overloaded by an institution that is happy to exploit them (and as a result, your group). Do not seek to assign blame in any contacts with the institute - this sets up a confrontational situation they will feel (right or wrong) they should defend to the hilt. This is why a representational body, which likely has people more experienced in resolving these issues, is so important.
You are not going to war, you are seeking a solution or solutions. That will almost certainly involve compromise.