Hello Vagtuning!
Okay, I know about suction nozzle pumps and, in particular, the problems with them, specifically for the BMW E39. However, if it's the same issue, I think it should function the same way. It can't be that two different, known patents have the same name for the same thing. Let's see how much I can help you with my knowledge...
If a Quattro has a full tank and is being emptied, which side of the tank must be emptied first?
The left one, where only the suction pump is located, or the right one, where the tank internal pump is located?
If the suction pump(s) are functioning correctly, both tank halves must be emptied evenly, otherwise the principle will not work. The principle that I am familiar with from my everyday life, which is used in BMWs, works as follows: on the right side of the tank, there is an electric pump, which is located in a so-called 'splash tank' and must always be filled with fuel. From this, a line runs through the tank to the left side, where it leads to the front of the engine. On this outlet, there is a pressure-regulating valve on the inside of the tank, which controls the pressure for the suction pump. Now comes the interesting part... The suction pump is, in principle, nothing more than another valve on the bottom of the left tank half. This valve functions according to the principle of a spray gun or sandblaster (a silly comparison, I know, but that's how it is...). The fuel that is directed there by the pressure-regulating valve flows past this pump valve at a certain, fixed pressure and, due to the resulting suction, draws additional fuel into the line. This then returns to the right side, back into the spray tank. In the bottom of the spray tank is a second suction nozzle pump. This also directs the fuel that comes from the left tank half past it. Same principle applies: due to the remaining pressure, a suction also occurs at the valve, which draws fuel from the right fuel tank half. Subsequently, this line ends in the suction chamber of the fuel pump. Therefore, if one of the two suction pump valves is defective, this side of the tank will not be completely emptied, and the vehicle will remain with an apparently half-full tank. The problem with the BMW is that the suction pump for the left fuel tank half must be purchased separately, around 12 Taler in this case. However, if there is a defect on the right side, the entire tank must be replaced, as the valve, fuel pump, and tank are a fixed unit... this is very expensive and time-consuming!
The suction pump is operated using two different pipe diameters.
If the right side says 20 liters, the left side (suction pump) is 10, then the internal pump pumps 10 liters to the left and receives 15 liters as a return flow, which it then gradually forwards.
But a Quattro, where there are 10 liters on the left side with the suction pump and 0 liters on the right side with the internal pump, cannot run (or start) because it has no way to access the fuel that is located on the left side in the chamber.
So, a fuel pump that specifically pumps fuel from one chamber to another, I only know from airplanes...

If it's as you describe, then the suction-based pump on the left side must be defective, and the tank on the right side must be completely empty, including the fuel pickup. Maybe ask the dealer what individual parts are available for this suction system for the Audi, and specifically replace the parts that are related to the suction pump on the left side. Then both halves of the tank would have to be emptied evenly again.
Greetings,
Dominique