If the OP isn't bothered about a reservation, selecting an off peak train on that day that has no Advance quota left, if there is one, would do it. I suspect however TCBC is being used for the seat selector, otherwise the answer is "use any other booking site"?
Many booking sites don't offer these fares (or at least didn't the last time I checked), but yes if the aim is to use the TCBC seat selector, then the solution may be to book directly with Avanti and then tweet them to request a change of seat reservation.
Initially Avanti were not changing seat reservations for people who tweeted them but they had to relent when they realised they were losing a lot of custom to Trainsplit/TCBC. When I last looked, Avanti were happy to do this, though some people did report that it could take a few days and needed a bit of chasing in some cases.
By the way, to expand upon the jargon in the opening post, I will clarify that SVH is a Super Off Peak Single (online) that can only be sold in conjunction with another Single ticket (of any type) in the other direction; SVH stands for "Saver Half" however the old Saver Half name was deemed too complicated, so it was changed at fares 'Simplification' in 2008. I will edit the opening post to comply with forum rules.