I'd also suggest checking that your fuel tank is being vented properly.
I mention this because it sounds like the car will restart on its own if left overnight... which in my mind points *away* from it being an IP issue (since an IP is unlikely going to heal itself, repeatedly) and more towards something that can resolve itself, given enough time. Venting is one example.
Next time it quits you could try opening the fuel cap to see if you hear a rush of air and can then get a re-start.
The VW fuel caps are generally pretty reliable, but you never know, and if someone put an aftermarket cap on at some point ya really never know.
Another possibility would be the crank sensor, which has been reported to sometimes develop a failure mode where it's fine when the engine is cold but degrades over time as the engine is running and things heat up. I'd expect a code in that case, but the ECU often misses things, particularly if the engine is in the middle of dying.