Exercise where lagrangian is needed?
I think a situation in which corner solutions may occur would be more fruitful. Cobb-Douglass ensures an interior solution, so they can use the expenditure share trick pretty easily, for any number of goods. Perhaps a really simple CES function, like $x^.5+y^.5+z^{.5}$? Or, if you still want an interior solution, pick something that's not a well-known function, but where marginal utility becomes infinite as either good approaches 0, like $\ln(x) - \frac{1}{\sqrt{y}} - \frac{1}{z}$ You'll still get tidy derivatives, but substitution won't be possible.