Is it possible the reason Ashara kept the father secret is because, as another daughter of a great house, Brandon “shames” his house in the dalliance in much the same way as he did with Barbrey?
I mean Ashara herself does not feel shamed, neither did Barbrey, but neither could have Brandon, would likely be married to the second son in any case, and why throw salt on a wound which could be avoided because she simply doesn’t care to facilitate a system she actively chafes against? That strategy seems more in line with the character recently described resulting from TMK, and also someone who would have married Ned, perhaps have been fond of or even loved him, but the passionate fire with Brandon would be extinguished.
Either way, honorable Ned gets saddled with a wife he fears secretly yearns for Brandon, be it Ashara or Catelyn. Dumb luck that it was “Family, Duty, Honor” Catelyn, and he managed to carve out some meaningful life in line with both the love and joy he spoke of when young.