Sorry for the late replies guys :<
Well, it's not something that I would like add to the litetween behaviour, if I add more and more features into it, the behavior will slowly be infested with more bugs and the interface and actions will become complicated. For things that can be done with event sheet, I prefer it to be done on event sheet. These kinds of things like controlling the tween or randomizing it really should be done with event sheet.
Try setting it on 'Enforce' instead of compromise, it looked like that is what you need. Compromise will try to 'compromise' with another behavior that change the object's position, for example if you have sine and tween run together, it will try to still play both instead of trying to compete with the behavior for controlling the object position.
On the Litetween start there's an option to choose 'start from last recorded' or 'start from current' which will start the tween from the proper position. Litetween is originally made to make movements of GUI, so it is simpler to make it defaulting to 'start from last recorded' instead of current. But if this is not what you mean, please let me know and I'll check it again.