As you mentioned yourself, this is a difficult situation. There are a lot to say about it but I'll try to advise on some of the main points I see.
You say you are new to the company. In two months you probably haven't gotten the full hang of things. And you now also have this situation you are asking about with conflict between team mates. Most likely, you don't have enough information about the full picture, so you need to address that first.
As a new manager you probably had one-to-one interviews with every member of your new team in order to know them better. If you haven't, you should. You can find a lot of resources online about conducting such discussions. Besides, as a manager, you will probably have to have such meetings quite often unless you want to evaluate people's performance only on an yearly basis, as it happens in many companies.
Speaking of which, you should talk to HR and ask about the existing performance evaluation process and see what they are using in the company. It might be something good, bad, or anything in between. No matter what it is, you need to know about it. You don't say what kind of a manager you are and what your exact responsibilities are, but if you are their line manager then you can probably also ask to see everyone's CVs and previous performance evaluation reports. You will be doing these sort of things as a manager going forward anyway, so you might just as well start with some context, not from scratch.
Once you get more information, you can start painting a bigger picture of what's going on, and see if the reviews seem indeed to be unfair and biased as the individual says. I have some doubts though, because if they were unfair and biased then there would be more people in this situation, and you also said "which has also led to switching teams within the company". Unfair and biased would have probably ended up with a firing, not a transfer. But this in itself is also a red flag, because people within the company might not be willing to fix a problem they just toss it around to others to deal with.
It's probably obvious already, but I will point it out just to be safe: don't mention to HR that you are asking for the performance review process, previous reports and CVs because of this one individual. You are asking because it's your new role and you just need to know about it.
You need to do your own evaluation of the individual (in fact, for everyone going forward). As a manager you can properly evaluate someone's performance only if you can do their job (many managers can't perform a subordinate's job, and from there all sorts of issues). You said "(software development) team member" so you need to understand that. Looking at your StackExchange profile though it seems you have this thing covered already, so look over their work and figure out if the job matches what you would expect from someone in that development role.
It's true that you should try to look at the entire team's performance and focus on the whole team working optimally, instead of trying to squeeze top performance from each individual and hoping that that somehow proportionally reflects in the whole team afterwards. But that's not an absolute. In team output an under-performer might not always be an issue, but teams are composed of people who have emotions and can become resentful if they are working and putting in effort and someone else isn't doing the same. They will carry that person along for a while, but not forever.
So whenever someone isn't pulling their own weight that's an issue that needs to be addressed. The classical response to that is the Performance Improvement Plan (PIP). Just make sure that it's indeed a problem with the individual and the solution you chose is something that the individual can fix by themselves. If the problem is some place else (e.g. maybe they were hired by mistake in a role the individual can't really fill, or something else) then obviously you can't ask the individual to fix it.
The PIP will be the long term approach (for everyone in this situation) but right now you seem to have a conflict on your hand between this individual and another team mate, conflict which needs to be resolved. As a manager you should first try to mediate the conflict, and help the two parties figure it out together by focusing them on whatever issue triggered it (otherwise people can move from the issue itself to the person and start attack each other on other levels). If things don't seem to go anywhere then as a manager it will be your responsibility to make a decision and put an end to the conflict. Think carefully, try to be fair and choose wisely. Support your decision with as much information as you can gather.
Finally, remember that you are a manager now. Your role is different than that of a software developer, as you've been so far. You will have to take this role seriously, dedicate the proper amount of time to it, and learn how to manage other people and situations. You have to put in effort at learning management just like you did learning technologies and programming languages. Managing people is harder than writing code.
You can of course involve everyone in your decisions, or help improve your decisions by asking input from everyone else, like rate each other's performance anonymously on a number of criteria as you said, or choosing those criteria together (something that everyone finds fair and unbiased), but at the end of the day you might have to take a decision just by yourself, one that might not be on everyone's liking.