Yep, insurers know about the factory fitted immobiliser, the car (2004 Vauxhall Agila) can only be started with the key. I believe the battery operated remote is purely for the convenience of locking all doors at once - I can do that by the simple expedient of quarter turning the key in the lock twice.
My 2004 Vauxhall Signum, should be similar tech to your Agila I reckon, doesn't do the 'quarter turn unlock' business if I've locked it like that with the key. It fires off TWO alarms, and won't start at all!
I had to go the whole hog, buy the new key AND have it 'connected' to the car's systems by the VM dealer.