How would I reduce the number of clicks required to answer this LG phone handsfree in the car?
Time for the "standard-reply": tasker. I will not go so far as to give you a complete "profile" (as I don't use that exact combination), but at least a raw idea (as for general info on what Tasker is, please see its tag-wiki).


Tasker understands the required triggers, so you can use a combination of e.g.


power connected
proximity sensor
bluetooth connected
voice-control
call state (e.g. "incoming call" or even SMS)

Tasker can deal with the necessary actions, and for many more things there are additional plug-ins available. It supports e.g. (without plugins)


call divert (even restricted to certain numbers)
enable car mode
load or kill an app
volume control
popup task (e.g. with 3 buttons: accept, reject, message)
send SMS



Much more is possible. So you could setup your very special and personalized solution – not only for car-mode, but for many other situations. And while root would give you additional possibilities, it's not required for the described purpose (as far as I can tell).