The word "trumping," which seems to have become popular lately, is a metaphor. In certain card games, notably in the game of contract bridge, a suit may become "trumps" and a card in that suit will be able to take a trick even if it is of a lower rank than one or more of the other cards being played in a different suit. For instance, if the "declarer" takes the bid at four spades, then spades are trumps. All players are required to play the suit that is led, if they have any cards in that suit, but they can play a trump card if they don't have any cards in the suit led. So a card in the trump suit can take the trick even if one of the other players has played a higher card in a different suit. The word "trumping" derives from card games, and the metaphorical meaning is simply that it overrides everythiing else.
There are other card games in which trumps do not have to be established by the bidding. One of the players, probably the dealer, may simply specify that a certain suit is trumps.
In Alexander Pope's "The Rape of the Lock"  there is a famous line quoting the heroine at the beginning of a game like whist:
"Let spades be trumps! she said, and trumps they were."
The meaning of "trumping other considerations" is that the honour of the regiment overrides, or is of superior value to, all other considerations.