For me the discussion is purely academic, but if I were designing the interface for a DS version of LCS, I'd completely cast out the current UI and use both stylus and buttons, redundantly whenever possible -- in my mind, the interface would be ideally designed if you can set the DS on a table and prod it with a stylus OR hold it in both hands and press buttons to play the entire game with minimal required crossover between the two styles, just player preference.
While exploring the map, the top screen would hold character status information and the bottom would hold the map. Holding the stylus on the map screen could move you in that direction. Tap doors to open or unlock; if there ought to be a confirmation, so that the player doesn't accidentally take risky actions, pop up a radial menu (as in The Sims) or a drop-down menu, and let you tap the option to take it. There would be a button to pull down an action menu or use things in the corner, and tapping options in the menu would select them. Your actual DS buttons would also be available to serve the same functions -- L and R could be used to drop down an action menu, and arrows would navigate the menu.
I'm not intimately familiar with the specs for the DS, but from what I've seen I'm sure it's plenty powerful to handle the processing involved in LCS. The perception that LCS uses complex simulation of the issues mostly comes from the ungainly lag when lots of time passes, and there it's not the simulation that's expensive, it's the sheer amount of simulating that has to be done. A more polished graphical game would hide this by making a pretty animation of a calendar tearing away or scenes from prison or something, and you'd not think anything of it, since you'd recognize that it's just "a long time passing".