My k03 GTI is quite rough on a cold start. And like yours, idles nicely once it's warmed up.
I'll be keeping an eye on your update.
I'm also getting misfires/fuel cuts, or both, under load, but I think this might be the standard fuel pump struggling (only started with the problems under load after I got the map revised from Stage 1 to Stage 2) so this could be unrelated to the bad cold starts in my case.
Fingers crossed the new injector sorts it for you anyway.