Your premise is wrong ... re-installing does not change or delete the existing catalog.
Furthermore, you can just enter your serial number into the trial version, Help->Lightroom Registration, and its not a trial any more ... very simple.
Unless I am missing something glaringly obvious, I cannot find any option to buy, register, upgrade to a full version of LR. I can see I have two days left of a trial but there is no 'Lightroom Registration' option under help to just simply progress to purchase.
I also find it confusing and frustrating that options are wrapped up in CC membership - I own CS6, don't want to pay monthly - I just want to buy it, be assured I have a full version and get on with working with LR as I have been.
Any help would be greatly appreciated - I've exhausted forums and Adobe website. As I say, might be missing something simple - so humble enough to have it pointed out to me.
It's been a while since I converted a trial LR to a fully paid version but I seem to recall that each time you get a dialog box that tells you how long you have left of the trial and also gives the option to buy. I'm not sure what the CC trial version looks like during the trial, however, so it may be that you have this.
All the way at the bottom, right hand corner - LR standalone.
if you have CC version installed now, remove it first. Uninstall does not touch the catalog file.