I use a non-free application with exactly that feature:
VanDyke.com > Products > SecureCRT
It support left-button select or Ctrl-Shift-C (to clipboard), and right-button or Ctrl-Shift-V (to paste). I use either, depending on how keyboard- or mouse- centric the task is.
SecureCRT also supports pre-configured login (scripts), multiple sessions (tabs/windows), and full scripting (extensibility). I find that when doing a lot of remote support, across multiple sites, these extra features are critical.
I have used it for years on Windows (or under Ubuntu + Wine), but it was ported to Linux in early 2011. I have not yet found a free alternate with all the features it offers.