I think that your dragon, lady, knight idea sounds very original and fun. You may consider allowing all the pieces to have special abilites or bonuses. If I were you I would start with the d -l -k idea but expand so more advanced players of your game could get into it and have something to work for (like that +6 dragon of ass-whooping)

Here is a simple way to add a little complexity to that Dragon, Lady, and Knight (RPS) idea...while still keeping it simple

As mentioned earlyer...have a map...a nice little map divided into a grid of 8 by 8 (or whatever)...each map cell has a R,P, or S flag (sorry useing the RPS name for now)...and each character occupies one map cell...say you have a knight fighting a Ogre on ajacent map cells...the knight is standing on a R map cell...the Ogre on a S...the player in control of the knight choses the Rock...the Ogre chooses the S...knight wins and does one point of damage...PLUS one point of damage for standing on a R map cell...and as the Ogre was standing on a S (which loses toa R) they take an additional point of damage)...in addition to all that, this causes the map cell the Ogre is standing on to become a "Rock" as well...

Then as the game develops and the players "raise levels" (or whatever)...not only is the map cell directly under the player counted...but ajacent ones as well...say the player is standing on a R and is at level 4...if there are four other Rs in nearby map cells connected to that same cell then thouse count tword damage as well (... character gets a point of damage for each level they have obtained, which is just for winning the RPS round ....plus all the map cells connected to the same map cell they are standing on add a point of damage too (if it is of the same type as what they chose in the RPS round (up to their level number)...and if the creature they are fighting choses the same RPS choice as the map cell they are occupying then the creature takes additional damage from that cell (and all connected to it up to their level number...and those same map cells become the type the player won the RPS round with)...

if the player is on a S and choses P...the monster he is fighting is on a R and chooses R...player wins "Papaer beats Rock" and the monster takes players level damage, plus damage from the map cell (and all like flagged cells next to it up to the monsters level number)-(this is because the players RPS choice beats the map cell flaged type)...the players land then changes to a R (as a R beats a S)...basicly the land types resolve themselves useing the same RPS rule set...

In the case of a draw (both parties chose the same RPS choice) then both map cells under the player and monster (includeing those connected of the same type up to each sides level rateing)....all these cells become the RPS type that beats the RPS type chosen by the player and monster...

adds some stratigy to the proceedings...

as for gaining levels doing this...could be just as simple...start on level 0 (zero)...and if you survive a battle without takeing damage...you get a "level point"...if you collect one more level points over your current level rateing then you increase a level...start at level zero and get a level point...you are now level 1...at level 12 and collect 13 level points...you are now level 13 ...such level points could be awarded for other things as well...such as winning a RPS round while standing on the same map cell type you chose (and haveing enough like type cells connected to it as equel to your current level number)...losts of interesting gameplay mechanics possabilities could be explored given this frame work