The charge only applied if there is a change customer address first line such as the building number and road krub. Or if there is a move of the license from one customer number to another. Right now you currently have 2 customer numbers.
For change of PC, there is no charge for reinstallation krub. Please kindly note down the terminal Serial Number (S/N) at the bottom left of any Bloomberg software window as you will need that serial for the reinstallation into a new PC. And yes you can download the software from "bloombergsoftware.com" -> New Installation link krub.
Kindly let me know if all is well when you try or if you need further assistance on this anytime krub.