We have a TT with the TFSI engine and it splutters a bit to start when cold. We never got to the bottom of it and we have had the car for nearly 70k miles. I just keep the car turning over for a little bit longer. Never an issue when warm. I remember a post from Nikki Gower who runs RTech about it a few years ago who I think referred to a valve that starts to leak causing initial low fuel pressure on start up. Worth giving them a call but they are based in Nuneaton.
However, Will who runs VRS is very clued up too and so should be able to help.
However, you have APR software and I might be looking there. I have the same stage 2+ On my S3 8P and I kept being told it would adapt as I changed and improved hardware.
I had the software fitted at the same time as having a APR downpipe and a Nuespeed intake.
I then changed the downpipe for a full BCS system with a 100 cell cat, a Revo intake, a Wagner intercooler and fitted an RS4 pressure relief valve. No alteration or tweak to the software when my car was at Awesome for other work. A custom map would have taken account of these changes. There is specific APR software that can be added for the runner flap delete but you need to check that it really has been done but as mentioned, the software is a bit too generic and cannot be tweaked for the particular hardware on the car.
If I was starting afresh I would always have a custom map from somewhere like RTech but I think your first port of call must be VRS as they provided the software.
Awesome are quite helpful and you could call them as well to see what software has been recorded on the APR database for your car.