If you migrate your clients to another server and IP of new server changes then you need to update IP in your brand on Site.pro licenses page from old value to new value after migration. The plugin re-installation or other actions are not needed.
Note for On-Premises Brand: The IP update on licenses page is needed only if you migrate hosting server with clients. In case if only builder server is moved then no actions are needed at all. There is no need in having IP for builder server on licenses page if that server is used only as builder and not as hosting server with clients.