CC is licensed by logging into your Adobe account with the AdobeID you purchased the software with. Uninstalling doesn't matter as much as never logging into CC on the oldest system. You cannot use PS-CC+LR on one computer, PS-CC on a second computer and LR-CC on a third computer. Your AdobeID-CC login can only be on two computers and then what part of CC you use on either is up to you.
What happened for me is that I had PS+LR installed on two computers, then replaced the older of the two with a new computer. When I tried to install and login to CC on the third computer, it asked me to deactivate the other two, which I did. Then I only ever logged into CC on one of the other computers which reactivate it on that one, then I had two activations again, but couldn't login to the oldest computer, which I didn't want to do, anyway, since it was old and slow.
From what it sounds like you have a Main and a Backup and a Laptop. That is three computers. You have to decide which two of those three systems you want to use PS+LR-CC on and logoff the other one and probably set it to not login when you start your computer.
Or are you saying you have PS+LR-CC on two systems, and you also purchased a LR6-upgrade-serial-number and have a LR5-serial-number as well and are having trouble installing the LR6 upgrade on the third computer, and somehow the CC install on the two computers is blocking the serial-number install on the third?
Be specific as to which versions (CC vs serial) you have (or ever had) of which products on which computers and what you're trying to install that isn't working.
On the computer you installed the stand alone version, the one you have a serial number for. Open LR and Sign Out of your Creative Cloud account. Make a New Adobe Account, using an alternative email address not associated in any way with your CC account, and sign in with that account. You will then be able to enter the serial number and activate LR 6 on that computer.
To complete all of this you might need to sign out of the CC desktop app and LR on one of the other computer first.