I've only been a BT TV customer a short period of time as well (switching from Sky) but I'm pretty confident that the answer is no. On BT TV (and Freeview generally) On Demand is On Demand i.e. you don't download it ever, you just stream it when you want it.
The only negative I can see to this is if you take a while - which I occasionally do - to watch a series and it then gets removed from the On Demand service. I suspect this is unlikely on the BTTV Player part of the service but perhaps more likely on the other catchup services.
I understand but unfortunately you just can't do it. You have to look at the BT TV Player part a bit like Netflix i.e. it's just an internet streaming service.
You might also want to browse on YouTube for Freeview Next Generation which will show you what the new interface (that I believe will come to BT boxes early in new year) will look like. Unfortunately that wont add the functionality you want either but it does deliver a more modern interface (all be it I'm sure there will be those prefer the current one).
The above said if you take a look at the new Sky Q interface that is going in a similar direction (although you can still download from the On Demand services)