IAT has a big impact on the MAF accuracy and the engine fuel control.
Some engines have a 3 wire MAF with the IAT in the intake between #3 & #4 intake runner. Not sure if the E53 was like some of the other cars with later versions having a 5 wire MAF with the IAT as part of the MAF.
MAF reading also sounds very high, over reporting, which will cause more fuel to be added to the engine. Most 3.0l will have a MAF reading of 3.6 g/s in park/neutral @ 700 RPM.
If your MAF reading is really in the 7+ g/s range, you have a problem, but also make sure the IAT problem is not impacting the MAF reading/conversion. IAT readings also impact ignition timing as well. I would bet your LTFT are in the negative range, which they should not be.
Suggest you consider the OBDFusion App, $1 cheaper than Torque and not at buggy. OBDFusion along with the VeePeak BT interface from Amazon is less than $20. OBDFusion is also available for iProducts, the App and interface for an iProduct platform is typically under $30. iProducts typically require a Wifi interface rather than a BT interface.
You can also use the App to Log the O2 sensors and this will show you what kind of health the catalytic converters are in, you may have already done damage and caused them to melt down. Typically you need an exhaust back pressure tester, although a compression tester may work, it usually does not do a very good job at low pressure reading. You can get a proper tester for around $37 -
Amazon.com: Tool Aid 33600 Exhaust Back Pressure Tester: Home Improvement