Have another read through
@majeric's thread linked in post 16. He has done what you intend to do and all your questions are answered, albeit an experience group of one!
But someone else in this thread also confirmed the trailer module functions worked without need for coding, so now that's two saying it's plug and play.
People saying something structural will be missing are mistaken. Or rather they just don't realise the towbar, or hitch for you, is not just the detachable ball, but also the structural crossmember that the ball attaches to. There's no physical way to fit the ball to your car and be 'missing something structural' since you must fit the crossmember! Note that in the UK/Europe the OEM hardware is a vertical detachable ball setup made by Westfalia. You will have a similar crossmember that likely provides a standard US square receiver, I would guess. So not same part number as majeric used.
So seems like mechanically and electrically it'll all work out fine. However, big however, sounds like you want to actually tow and not just fit a bike rack, which on paper isnt permissable with a Model 3 not factory equipped for towing. Well that's up to you! As above it will physically work, but if anything happens you could be in hot water.
But I have to ask - why all the trouble? I heard from a friend that in the land of dreams you just pick up a Ford F150 for 100 bucks and it can tow your house.