Ah yes, those complicated and annoying minimum fares, that lead to all sorts of anomalies!
Cheap Day tickets are indeed £8 minimum, as are Savers. However, Standard Day/Standard Open return fares are £16 minimum. You'll struggle to find many SOR costing less than £24 anyway, given that they are long distance ticket type. The SDR restriction is probably to stop young commuters getting a cheaper deal than a season ticket.
Nationalrail.co.uk will not sell you any tickets, so there is no point using it (IMO). If thetrainline offers such a ticket, purchase it and bring confirmation details in case you are challenged (in reality, you won't be), as the TOCs have then entered into a contract which they cannot legally refuse to fulfil at that price.
But shoving incorrect details off NRES in front of a ticket seller is unlikely to hold water
BTW it's worth mentioning that the minimum fares are abolished during July/August, bear that in mind for next Summer!
I paid £9 or whatever it was to get to Leeds the first time I did that journey at peak time, then I thought of a way round it and asked for a ticket to Hebden Bridge (about twice the distance) instead, this allowed me to get a Saver and only pay £8.

I always got away with using a fully discounted CDR ticket on the 09:43 Virgin HST to Leeds though!