'The precedent for the Ajax-Pieces not being able to capture with their adopted Commoner moves is the Pawn.'
Not really, the Pawn uses one type of direction in which it can move without capturing, and one type in which it can capture, but none in which it can do both. Likewise the Yeoman, Steward, and other offshoots.
All the traditional 'crownings' of linepieces (Shogi, Duke of Rutland, Wellisch hex &c) include the ability to capture with the extra move. Indeed you use images whose usual meaning is a piece that can capture in all its directions. Your new images could prove more popular for straightforward Rook+Knight+Ferz and Bishop+Knight+Wazir. As it happens I have been writing a page whose introduction mentions Rook+Knight+Ferz, although as far as I know it has yet to make it into any actual games.