Dear librick,

Hope you're well. As this failure seems heat-related - car runs well, once "warmed-up", but runs poorly when cold - I think you should check:
(a) the engine coolant temperature sensor (on the driver's side of the head, towards the back (North American models) and
(b) the ignition power stage. This is mounted on the inner fender wall, just behind the headlight. On a turbo-equipped engine, it may be on the passenger-side fender of the engine bay (normally-aspirated, on driver's side of the engine bay). It has a single electrical connector, is mounted on an aluminum base, and has a pale gray "cap".

The ignition power stage more often fails when hot, but I recall seeing posts, when failure occurred when the engine was cold.

Neither of these is a super-costly item. A failed engine coolant temp sensor usually will set a code (1-2-3).

Hope this helps.

Yours faithfully,

Spook