Does Manaroo's ability cause it's TL to be assigned to a friendly ship if it already has one assigned to a different ship? In other words, does the player have a choice of whether to accept Manaroo's TL? If the answer is no, what happens if Manaroo has weapons engineer?

Manaroo does not have the option to trade a partial number of tokens, so if the friendly ship is handed a new blue target lock token and it is already holding one, it can choose which of the two to discard (the old or the new). This is similar to the situation where Manaroo has weapons engineer and transfers two blue target lock tokens - the friendly ship (unless it also has a weapons engineer) cannot maintain two target locks and must choose one of the two to immediately discard. Technically both of your questions put the friendly ship receiving the target lock in the same situation.

So you actually have a lot of flexibility here! Well, sorta.