All fares comparisons 'must be made using the same ticket type', so says the Routeing Guide.
Sonic2009's fare is a Super Off Peak Single (SSS), Any Permitted. There is no Super Off Peak Single, Any Permitted from Barnt Green....
The Off Peak Single, Any Permitted (SVS), is less from Ashchurch than it is from Barnt Green, so that would fail the fares check rule.
The Anytime Single, Any Permitted (SOS), is more from Ashchurch than it is from Barnt Green, so that would pass the fares check rule.
I'd conclude that on if travelling on the Anytime Single, Any Permitted, you would, according to the Routeing Guide, be allowed to travel via Birmingham from Ashchurch to London Terminals. Less clear cut if travelling on the Super Off Peak Single. But, if as bb21 says, any of the fares can be used for a fares check, then it is allowed.
Finally, and despite the Routeing Guide, if a booking engine gives up an itinerary via Brum for the Super Off Peak Single, then that's evidence of a contract term and travel should be allowed regardless.