IMHO there is very little difference between a private sale in autotrader and the auction "trade price" + buyers indemnity (which is a lot more for the private punter than it is for the trade at BCA), especially for the type of car you are looking for.
However, an advantage for buying from auction is that if a car is on the HPI register, you will be notified before the sale starts - but you have to listen out for it.
There is hardley ever a real bargain at auction. A few out of an entire days sale maybe.
If its mega cheap and the traders aren't bidding, then from experience there is generally a reason for it.
If its the right car with the right spec, right colour and good credentials, it will fetch massive money - I've known the trade to pay £1000+ into book. ie £1000 more than trade value, just because it was a nice example and had everything going for it, knowing they could be cheeky with the asking price on the pitch (forecourt).
Many traders scour the autotrader and local rag too.
In short - if you want choice, then go to an auction. If you can bide your time and have patience to scour the ads and drive around looking at cars then buy private. If you buy private then HPI it over the phone there and then, before you part with a single penny.