Having team-members split the grade among them by their personal contribution

A problem I see here is that this scheme may motivate people to divide the points "tactically". Say our group project is worth 10 points and I only need 5 for my goal (which may be the least passing grade or the best grade or whatever). Then of course I would take only 5 points and give 15 to my collegue (which is more than the project is worth).

Also, it might motivate people to look for their partners tactically: If I am somebody who does not trust other people and wants to do everything themselves (something which should ideally be discouraged in group projects), I choose the person who cares least about their grades as a partner and get much more points than my project is worth.

Moreover, with this setting, you treat the grades as some currency. students will find it okay to do less/more work because they can themselves discuss and haggle about the grade they got. IMO this is also not something which should be encouraged.

I like this as an experiment in ethics, but not as an actual grading scheme.

If you allow team work, you will have a small number of people getting better grades than they might deserve. And...so what? It’s not intrinsically different from making homework part of the grade, where friends will help each other. Your job is to teach and assign grades you deem reasonably, not perfectly just.

That is to say, I’m no fan of any such scheme involving students “dividing the pie,” your self-declared research interest. I for one would have neither felt comfortable nor able to write meaningful evaluations for all my class members, as was suggested here too. Some people might stand out, and you might enjoy sharing that with your teacher; but I wouldn’t have enjoyed pointing fingers at those struggling. And for many I’d have no true opinion. That is not even to talk about personal feelings about peers almost inevitably shading a student-given grade.

I don’t think the benefit of being marginally more just makes up for the hassle and risk for trouble such ideas involve.

This isn't the sort of thing you should introduce after the fact. If you make it part of the course design, known to students at the start, then it might work, though it might just cause more complaining from the students. Teams can "share" the work while doing very different things. Each can consider their own contribution to be essential, while their teammate(s) consider their own to be more important.

But changing the grading structure of the course midstream to the potential disadvantage of some students is questionable.

But, my answer to your earlier question also covers this sort of situation: Peer Evaluation. It lets you learn about some things, but retain control over outcomes.

But repeated questions on the same issue implies that you need to rethink your course design.