Hi Kenan,
The point here is to reinstall the SQL Server cluster without downtime. As you have pointed, to install a new SQL Server cluster is required new IP address, DNS Name. In this case, you may have a try to use CNAME to redirect the client to the new SQL Server
cluster instance. Before the redirection, you need to synchronize the data from old server to new server before. The threads below may be helpful to you on this issue:
CNAMES and SQL Clustering on Windows 2008
Defining CNAME on DNS for named SQL Instances
To make things simple, you may have a consideration to remove the node which is unhealthy, and add it back after reinstall the node.
Stephanie Lv
TechNet Community Support
