You mentioned that the modification to the motorbike was purchased at a Moto GP and not legal in the UK. It probably wasn't legal on the Continent, either (not everything that's on sale is legal). But that's not the point. The insurers deemed, and quite rightly so, that the bike was not roadworthy in the modified condition. That they wouldn't pay up in the event of an accident is perfectly understandable. But whether tax has been paid or not has nothing to do with roadworthiness.
So it still comes down to the point, if the insurance company will confirm that the vehicle is covered while out of the country and untaxed, then there is nothing illegal with using the car there.
What you mention with Brits cashing in tax and then using the car abroad is actually done. I have seen UK cars without tax disks in Spain, although I don't know whether these cars may not have been insured there locally. Likewise, German residents who expect to be out of the country for long periods can officially exchange their normal number plates for export ones for which tax is only paid while they are still in Germany. They can then continue to use those vehicles on these export plates while abroad, although tax has expired in the meantime.