[Hitman]

From what I've heard (here in subsim and other sites on WW2 submarines) the Germans didn't have a TDC that could keep track of a targets position automatically 'internally'. The crew would have needed to feed it current data for an update of it's aiming, of which bearings would be most important.

Correct, the german TDC had no automatic position keeper, unlike their US and british counterparts, but it could update the solution following the movement of the optics. When the Kaleun or IWO asked the TDC operator to set it in "Lage folgend" (AOB following) mode, the TDC bearing dial was directly linked to the scope or UZO, and moving them updated the solution correspondly. When set to "Lage nicht folgend" (Not AOB following), a blue lamp lighted and the TDC accepted inputs in all its parameters, (Distance, AOB, Spread, target length...) plus it did no longer follow the optics. SH3 has therefore a very good representation of the german TDC except that our home computers calculate the solutions nearly instantly (See what I explain next) unlike what the real tool was able to do.

I like to set up the periscope direction in advance and wait for any location to pass through the aiming line

That's basically what was done in WW2 U-Boats, not just because of the convenience of being able to aim at different parts of the ship, but also because the TDC was an analogue mechanical calculator and it took some seconds for it to provide the proper solution (At that moment a lamp was lit to show the solution had been created) after a change in any of the inputs. Obviously simply moving the optics (bearing) allowed for faster update than if changing other inputs, but still it was considered better to shoot at a fixed, pre-chosen bearing with a fully calculated solution.

My main concern is now the validity of modifying my targetting solutions at the TDC (where someone is magically plotting the ships' locations in real time) to compensate for my currently inept estimates of range, AOB and speed.

The most relevant factor here is speed, AOB is important but not that much (And also depending on the relative position of target and attacker, f.e. a 80º AOB instead of a 90º one is not as important as a 20º to a 30º error). Distance to target will only matter if you do angled shots. The Gyro Angle indicator on your scope screen is not causally there: The closer you are to a zero GA when shooting the less will distance matter. See my post about convergence here to understand it better: http://www.subsim.com/radioroom/showthread.php?t=121744

Also, speed and AOB will be less relevant the shorter the distance to the target. German commanders went really close before shooting, always as close as possible (Less than 1000 metres) and as much perpendicular to the target as they could, and that minimized errors in calculating/estimating target data.