Unfortunately once the trial (including the trial extension) has expired, there's no way to get another one unless you either run it on another computer or wait until the next major Media Center version which will include a new, separate trial period.
It's done this way by JRiver, I presume, to prevent certain types of users from resetting the trial period which would be in essence pirating the application since you could endlessly reset the trial period. I personally have my doubts they'll change how this works, since doing so could allow that sort of thing to happen. So yeah, there's probably nothing that can be done about it (since uninstalling the app broke the trial extension and caused all trials to expire after reinstalling) and you might be stuck running the trial on another computer if you have one.