Can you claim an illegal move in Blitz after pressing your clock (but without making a move)? In Blitz under FIDE rules, Player A makes and completes an illegal move by pressing the clock. Player B states that Player A's move is illegal, does not make a move, but quickly presses his own clock, as if giving Player A the opportunity to correct the move as common in a standard non-Blitz game. Almost immediately, before any other action by either player, Player B realizes that the rules of Blitz apply, stops the clock, and claims the match.

According to Player A, the claim is now "in his time", so is no longer valid. Is this a correct interpretation of the rules?
This is correct. You cannot continue to play (press the clock) and claim a win in blitz. Furthermore in the new rules pressing the clock without making a move counts as an illegal move and so loses (if your opponent claims). This is really just another twist on the old error of taking your opponent's king when he leaves it en prise which again results in you losing for an illegal move instead of your opponent.

In other words, if your opponent makes an illegal move in blitz just claim the game. If you start doing other move-like things (touching a piece, pressing the clock) your claim goes away.

The problem in your case is the same, by the way. An illegal move is only punishable once it is completed by pressing the clock. When you pressed the clock you removed the evidence for the illegal move.