Using bidding for combat resolution

but what if the player's had a hand of influence cards, one card for each point of influence they have. And these cards interact in a rock-paper-scissors kind of way, but with probably 5 options instead of only 3. To make an influence attack, the attacker plays one of his influence cards. Then the defender has to play an influence card that counters the attacker's card. They go back and forth like this until one of them can't play anymore. They both lose the cards they played. For thematic reasons, the cards could be labeled things like Charisma, Leadership, Vision, Connections, Resources, etc.

It has the same back-and-forth feeling of a bid, but the outcome is less certain. The player with most cards might not win if he has too many duplicates, so they won't be able to be certain of victory just by outbidding the other player. But I don't know if that's something you want in your game or not.

"When each character clashes against each other, the winner is determined by adding the number on each piece to the value of a card. Each player secretly selects which card they want to play, then reveal simultaneously. These cards are discarded and out of play until you use up your whole hand (9 cards), so saving important cards for later matchups is important.
The Light side also has a 'Noble Sacrifice' card which kills both characters in the fight, while Dark has an 'Eye of Sauron' card which cancels out the Noble Sacrifice. Some fights are bluff and guess work to play Eye of Sauron when they play Noble Sacrifice, or a higher number than them if you think they'll go for a number too."

There are any number of special cards you could add onto this system: cards that cause a siege rather than an immediate resolution of the battle, cards that cause the value of the other card to be ignored, cards that cause a bloodless retreat or the opposite...