Ok, I had a long chat with a very helpful guy in Goodyear's technical team and I have an answer. I wasn't aware that BMW approved tyres are 'star marked'. Now whether this means that all Bridgestone RE05s are star marked or you specifically have to ask for them to be, I've no idea. Anyway, the Goodyears I fitted are not star marked. What this all means is that within the international standards on tyre construction a small margin of tolerance is allowed on tyre sizes and we're talking a few mm here. With star marked tyres the sizes are exact. The guy said that the software in all but the very newest BMWs can get upset when the tyre is not star marked and so, presumably, could be fractionally smaller or larger. As far as the software is concerned the tyres are not travelling at the correct speed and so it detects a skid situation.
The only solution, which is what I kinda expected, is to switch back to a star marked tyre or, to be safe, a Bridgestone RE05. The guy agreed that matching the fronts with the rear Goodyears should help to fool the software. He said that if other Z4 owners have done this successfully then I'm likely to be OK, but he couldn't guarantee it.
So there you go. Looks like I'll have to go with Plan B and switch the rears back to Bridgestone.
If anyone would like to buy some Eagle F1s with a about 100 miles on them then let me know!