I'd put in in your Classic II, which is the most limited of the two machines, true, but also the one with what most people would say has the most character. The Classic II will also run (a special version of) System 6, under which it runs very fast indeed.
Having said this, if you are trying/ wanting to run System 7, an LCIII will seem a revelation by comparison with a Classic II, because of processor and bus speed, as well as RAM factors. The LCIII is a surprisingly capable little machine. This website, for instance, is running on what is for all intents and purposes an LCIII. You'd never get that sort of speed out of a Classic II, which was a technologically compromised as well as a less expandable machine. Still, I'd say the Classic II is the more interesting artifact of the two....
Having said this, if you are trying/ wanting to run System 7, an LCIII will seem a revelation by comparison with a Classic II, because of processor and bus speed, as well as RAM factors. The LCIII is a surprisingly capable little machine. This website, for instance, is running on what is for all intents and purposes an LCIII. You'd never get that sort of speed out of a Classic II, which was a technologically compromised as well as a less expandable machine. Still, I'd say the Classic II is the more interesting artifact of the two....