Just an update on this so I can give back to this great forum. Apologies that there are probably technical anomalies in the post, because quite frankly I have no idea what I'm talking about. But it may help others.
I had one original working key (the case had taken a beating, the flipping mechanism was temperamental and the key ring part was missing). So I bought a couple of blank keys from here with a view to creating two "new" keys (these keys have transponder chips on the correct frequency etc):
https://www.ebay.co.uk/itm/235062525373
The keys are of good quality - a nice feel to them - not light and flimsy. Very similar to OEM.
Many people say that Timpsons can cut/program the keys, but they can ONLY programme the transponder chip and NOT the remote buttons. In other words, you could use the key to manually open your car via the key hole (setting the alarm on) and start it (turning the alarm off). But you won't get remote unlocking of the doors and boot etc.
....but I didn't want a half-working key. So I called Rob over at
https://www.carkeysmobile.co.uk/. He's got over 25 years experience cutting/programming keys.
He managed to programme the transponder and cut the key to start with. It wasn't trivial. He also had issues reading from the car (it's based on Fiat apparently), but somehow managed to extract the relevant information (and no, he didn't require the "key card" to do this) to program the transponder.
The remote buttons were tricky too. When his device asked him to start adding the keys for remote programming, he added the original key first which worked fine. It then prompted him to "add another key", so he added the first "new" key which also worked fine. It prompted again to "add another key", but when he tried to add the second "new" key it didn't acknowledge it. It turns out that both the new keys evidently emit the same "signal", so his device thought he was trying to add the same key twice.
He repeated the cycle again, adding only the original key and one "new" key. And afterwards he found that the second "new" key (albeit unprogrammed) sent the same signals to unlock/lock the car as the first "new" key! This is good, you might think? But you'd be wrong.
As the
@CatmanV2 has pointed out on numerous occasions, QPVs use a rolling code mechanism. So what this meant is that the two "new" keys were competing with each other since they were sending the same "signal" when locking/unlocking the car. And hence the new keys were becoming out of synch with the car.
However, since the "new" keys were sending a different "signal" to the car than the original key, it meant that we could use the original key and ONE of the new keys successfully together.
So in the end, the solution was to use ONE of the new keys and replace the old case on the original key with the new case on my second "new" key. Thankfully the little circuit board inside the original key fit perfectly into the new case (I've seen some people on forums hacking around to make it fit). But it now means I have two working, good quality keys, with full functionality.
Thanks to
@CatmanV2 for his advice throughout!