Connor.
Before buying any expensive, whether Audi or non-Audi, parts and trusting to luck that the problem will be solved, I would carry out some pressure tests to find where the problem really is.
A compression test should reveal if there is an internal engine leak, though one or more low cylinder pressures could be the result of worn rings, defective valves, or damaged head gasket. A pressure test of the oil cooler with water at mains pressure together with a valve and a dial pressure gauge will reveal if it has a hole.
Both tests can be done by the competent home mechanic with some ingenuity and cheap instruments.