these cars are mostly controlled by some sort of modules for each part. modules can be seen as small CPU controllers that have the task of making a particular part work with the main ecu brain in the car. the transmission control unit is one. it is different from the rear haldex unit controller and so on. so what i am going to suggest is a combo of maintenance and computer fixes to help fix your issue. it may take a few or one of these measures to work. but doing all will still be good as it may be a maintenance fix.
1. change out your battery. it might be weak or old at this point. yes, i know your car still starts up but your amp draw requirements during hard driving does drain the power a ton in just a few seconds. the alternator may not be able to replenish a battery that can not hold high enough amp currents. again refer back to the modules you have that i mentioned before. i personally recommend a lithium battery as they supply power more linearly than any lead acid battery. don't cheap out and buy a weaker or even a smaller battery. it wont last long as these cars, from stock are supplied with a very high amp hour battery. the S3 is supplied with an audi 12v 72AH lead acid battery. so if you buy below this, it will not work properly over time. as we all know batteries degrade over time. i see so many people cheap out and buy something low-end and wonder why the car has problems after 2 years or so. lithium is guaranteed for 10 years by most companies. call to verify before you buy. but make sure the 12v lithium has at least the same amp hours or more. doesnt hurt to have more. not many offers that size with higher AH for our cars.
2. change out the haldex fluid. since you are in there i also recommend the bevel box fluid change. the main part of this is the haldex screen cleaning. its could be clogged up. it could also be low on fluid. who knows? do it. fix it.
3. the most important part of all this i feel is after you do #2 above, you'll NEED to cycle the haldex pump with vcds. i did all these and this part solved my issue.
everyone should be doing this every 12 months. think of it like a computer and it needs solid power but mostly needs a reboot once in a while. for instance, if you didn't give you home PC the right amount of power through the supply you bought, it wouldn't power the CPU chip, video card, sound card, mother board, etc properly or it could just even mess it all up. i have seen that before.
so, if your battery is shit, you'll need to restart this process very often, but its much better to buy a good battery first. its not that good if you need to restart the CPU every few hours, am i right?